Mechanics' 
Pocket  Memoranda. 


7. ,C)t NfCAL  SUPPLY  COMPANY 


DRAWING  MATERIALS 
SCRANTON, 

i'H'.'i  - U’CVA  AND  NEWYORK 


/ 


Mechanics’  ) 
Pocket  Memoranda 


A CONVENIENT 


ENT 

3' 

ALL  INTE 


TBOOK 


FOITULL jpfyqps  INTERESTED  IN 

Mechanical  EnAperin^f  Steam  Engineering,  Electrical 
jyg,  Railroad  Engineering,  Hydraulic  - 
;ineering,  Bridge  Engineering,  Etc. 


Eng 


BY 

INTERNATIONAL  CORRESPONDENCE  SCHOOLS 

SCRANTON,  PA. 


7 th  Edition , 277th  Thousand , 18  th  Impression 


Scranton,  Pa. 

INTERNATIONAL  TEXTBOOK  COMPANY 


Copyright , 1893 , 1594,  1897 , 1898 r 1899 , 1900,  The 
Colliery  Engineer  Company 

Copyright , 1994,  Oy  International  Textbook  Company  N . 
Entered  at  Stationers'  Hall , London 


All  rights  reserved 


PRINTED  BY 

International  Textbook  Co. 
Scranton,  Pa. 


3256 


PREFACE. 


£ a o 


The  first  edition  (2,000  copies)  of  the  pocket- 
book  of  which  this  is  the  outcome  was  issued  in 
October,  1893,  in  the  form  of  a notebook  contain- 
ing 74  printed  pages,  with  about  the  same  number 
^of  blank  pages  for  memoranda,  whence  the  title 
Mechanics’  Pocket  Memoranda.  The  little  book 
proved  so  popular  that  a new  edition  ( 10,000  copies ) 
enlarged  to  110  pages  was  issued  8 months  latei*. 
In  June,  1897,  the  blank  pages  were  discarded,  the 
work  was  entirely  recast  and  enlarged  to  318  pages, 
and  the  edition  (third)  consisted  of  25,000  copies. 
Before  printing  the  fifth  edition  (March,  1898), 
a large  amount  of  matter  relating  especially  to 
Plumbing,  Heating,  and  Ventilation  and  the  Build- 
ing Trades  was  taken  out,  replaced  by  tables  of 
logarithms,  trigonometric  functions,  etc.,  together 
with  directions  for  using  them,  and  other  new 
matter,  the  result  being  to  confine  the  work  more 
particularly  to  the  different  branches  of  engineer- 
ing and  mechanics. 

It  has  been  the  aim  of  the  publishers,  from  the 
first,  to  present  to  the  public  a handbook  of  a size 
convenient  to  carry  in  the  coat  or  hip  pocket — a 
pocketbook  in  reality— which  wrould  contain  rules, 
formulas,  tables,  etc.  in  most  common  use  by 
iii 


IV 


PREFACE. 


engineers,  together  with  explanations  concerning 
them  and  practical  examples  illustrating  their  use. 
We  have  not  endeavored  to  produce  a condensed 
cyclopedia  of  engineering  or  of  any  branch  of  it, 
but  we  have  striven  to  anticipate  the  daily  wants 
of  the  user  and  to  give  him  the  information  sought 
in  the  manner  best  suited  to  his  needs.  Our  aim 
has  been  to  meet  the  necessities  not  only  of  the 
engineer  but  of  all  in  any  manner  interested  in 
engineering,  and  in  accomplishing  this  we  have 
selected  that  rule,  formula,  or  process  which  was, 
in  our  opinion,  best  adapted  to  the  circumstances 
of  the  case,  describing  it  fully,  giving  full  direc- 
tions how  and  when  to  use  it,  and  not  mention- 
ing other  methods  (when  such  were  available); 
in  other  words,  we  have  made  the  selection  instead 
of  leaving  the  choice  to  the  judgment  of  the  user, 
which  is  frequently  at  fault.  The  exceedingly 
large  sale  proves  that  the  idea  was  popular  and  has 
vindicated  our  judgment.  We  hope  that  succeed- 
ing editions  will  meet  and  merit  the  same  approval 
that  has  been  accorded  those  preceding. 

The  present  (seventh)  edition  contains  the  most 
convenient  table  of  powers,  roots,  and  reciprocals 
of  numbers  yet  printed.  This  table  was  arranged 
and  computed  by  us  and  will  be  of  great  use  to  all 
having  occasion  to  use  it. 

International  Correspondence  Schools, 

December  1 , 1903 . 


INDEX 


A. 

Page 

Absolute  pressures 26 

Alloys 19 

Alternating  system,  Size  of  wires  for 239 

Aluminum  and  copper,  Properties  of 250 

Ampere 230 

Angles  or  arcs,  Measures  of 2 

Annunciator  system 248 

Anode  of  an  electric  battery 263 

Arc  lamps,  Connections  for 253 

Arcs  or  angles,  Measures  of 2 

Areas  and  circumferences  of  circles 82-90 

Irregular 119 

of  circles,  Table  of 82 

Avoirdupois  weight 3 

B. 

Batteries,  Storage 267 

storage,  Regulation  of 267 

Various  chemical 263  -266 

Beams,  Bending  moments  in - 152 

Cantilever 152 

Deflection  of 152 

fixed  at  both  ends 152 

Simple.  152 

Bearing  of  a line 276 

of  a line,  Deduced 283 

Bearings  for  line  shafting,  Distance  apart  of 193 

Bell  wiring 241 

Belting 140 

Rope 209 

Belt  pulleys • 204 

Bending  moments  in  beams 152 

Birmingham  wire  gauge 249 

Blow,  Force  of  a 140 

v 


VI 


INDEX. 


Page 


Blueprint  paper,  To  make 175 

Blueprints 175 

Boilers 158 

(steam) 158 

(steam)  Foaming  and  priming  of  168 

(steam)  Horsepower  of 168 

(steam)  Inspection  and  care  of 162 

(steam)  Prevention  of  scale  in 164 

(steam)  To  develop  dome  of 158 

(steam)  To  develop  slope  sheet  of 159,  160 

Bolts  for  cylinder  heads 217 

for  steam  chests 217 

Standard  proportions  of 22 

Booster 267 

Brake,  Prony 260 

Bridge,  Wheatstone 270 

Briggs,  or  common,  logarithms 32 


c. 


Cables,  Carrying  capacity  of 

Chain 

Testing  of  (electrical) 

Calendar,  Perpetual 

Candlepower 

Capacity,  Measures  of 

of  cables 

Cathode  of  an  electric  battery 

Center  of  gravity 

Centrifugal  force 

Chain  cables,  Wrought-iron 

Chains  and  ropes 

Change  gears 

Characteristic  of  a logarithm *.  . . 

Chemical  treatment  of  feedwater 

Chimneys 

Formulas  for 

Table  of  sizes  of 

Chord  of  circle 

Circle,  Area  of 

Chord  of 

Circumference  of 

Segment  of 

Circles,  Tables  of  circumferences  and  areas  of 

Circuits,  Derived,  or  shunt 

Motor 

Size  of  wire  for  arc-light 

Circular  pitch,  Formula  for 

pitch,  Table  of 

rings,  Area  of 

rings,  Volume  of . . . 

Circumferences  and  areas  of  circles 


239 

14 

271 

327 

236 

6 

239 

263 

121 

121 

14 


33 

165 

170 

171 

172 
116 
113 
116 
113 

115 
82-90 

' 234 
253 
253 
228 
230 

116 
117 

82-90 


INDEX. 


vii 
Page 

Clearance,  Piston 216 

Coefficient  of  elasticity 152 

Coefficients  of  expansion 19 

Columns,  Formulas  for  strength  of 156 

Commutator,  Sparking  at 259 

Compass  surveying 276-279 

Compound-geared  lathes,  Screw  cutting  with 182 

pulley,  Formula  for 138 

Compression,  Table  of  ultimate  strength  for 151 

Conductivity,  Electrical 232 

Conductor,  Direction  of  motion  of 233 

Size  of 235-240' 

Cone,  Formulas  for 117 

Conical  frustum,  Formulas  for 117 

Connecting-rods 224 

Connections  for  dynamo-electric  machines 252 

Copper  and  aluminum,  Properties  of 250 

Corliss  engine  crank-shaft 223 

engine  cylinder 221 

Corrosion  of  boilers 163 

Cotters  for  connecting-rods 226 

Couplings,  Flange 194 

Flexible . : 195 

Proportions  of 195 

Shaft . 194 

Course  of  a line  in  surveying.  276 

Crank-shafts  for  Corliss  engines 223 

-shafts  for  high-speed  engines .' . . 223 

Cross-over  tracks 325 

Cube  root 105 

Cubes  and  squares 106 

Cubic  expansion,  Coefficient  of 19 

measure 2 

Current,  Rules  for  direction  of  electrical 232 

Strength  of 231 

Curves,  Deflection  angles  of 286 

Degree  of 286 

Elevation  of  railroad 311 

of  saturation * 256 

Tangent  distance  of 287 

To  lay  out  with  transit 288 

To  lay  out  without  transit 290 

Curving  of  rails 309 

Cylin  der  hea  ds 217 

heads,  Bolts  for 217 

Cylinders  for  Corliss  engines 221 

Formulas  for  strength  of 157 

Proportions  of 216-219 

Stuffingbox  for 228 

Surface  of 116 

Volume  of 116 


INDEX. 


viii 

D.  Page 


Decimals  of  a foot,  Equivalent  in  inches  of 91 

Declination  of  needle 277 

Deflected  line ' 281 

Deflection  of  beams 152 

Deflections,  Tangent  and  chord 297 

tangent  and  chord,  Formulas  for 297 

Density - 25 

Derived  circuits 234 

Designs  of  machine  details 192 

Development  of  boiler  dome 158 

of  boiler  slope  sheet 159 

Diagram,  Slide-valve 188 

Diametral  pitch,  Formula  for 229 

pitch,  Table  of 230 

Differential  pulley 138 

Division  by  logarithms 42 

Dome  of  boiler,  To  develop 158 

Double  movable  pulley 137 

Draft  of  chimneys,  Formulas  for 171 

Drills,  Speed  of  twist 177 

Dry  measure 3 

Duty  of  pumps 144 

Dynamo  design 254 

-electric  machines,  Connections  for 252 

machines 246 

wiring,  Underwriters’  rules  for 245 

Dynamos  and  motors 253 

Faults  of 258 

E. 

Earthwork,  Calculation  of 306 

Eccentric 227 

Efficiency,  Lamp 240 

Motor 253-260 

Elastic  limit,  Table  of 152 

Electric  gas  lighting 269 

motors,  Application  of 261 

Electricity. 230-275 

Electrodeposition — . ' 269 

Electrolyte  of  an  electric  battery 263 

Electromagnet,  Polarity  of 233 

Electromotive  force 230 

force,  Formula  for 254 

Elements,  Table  of  chemical 16 

Elevation  of  railroad  curves 311 

Ellipse,  Formulas  for 115 

Emery  wheels,  Speed  of 176 

Engine  horsepower,  Formula  for 185 

English  and  metric  measures,  Conversion  tables  of  7 


INDEX. 


ix 
Page 

Equivalent  decimal  parts  of  one  foot 91 

decimal  parts  of  one  inch 91 

Evolution  by  logarithms 46 

Table  method  of 103 

Exhaust  heating 173-174 

ports,  Dimensions  of 217 

Expansion,  Coefficients  of 19 

Exponents 32 

External  inspection  of  boilers 164 

F. 

Factors  of  safety,  Table  ot 151 

Prime 80 

Failure  of  dynamos 258 

Falling  bodies 120 

Feedwater  heaters 166 

Methods  of  purifying 165 

Testing  of 164 

Field  magnet 255 

magnet,  Reversal  of 258 

Filtration  of  feedwater 165 

Flange  coupling 194 

Flanges,  Pipe 215 

Flexible  coupling 195 

Flexure,  Ultimate  strength  of 151 

Flow  of  water  in  pipes 147 

Fluxes  for  soldering  and  welding 24 

Foaming  of  boilers 168 

Foot,  Decimals  of  a 91 

Force,  Formula  for  electromotive 254 

Magnetizing 255 

of  a blow 140 

Forces,  Resultant  of 137 

Formulas 93-302 

How  to  use 93 

Frog  (railroad  work) 312 

Angle 313 

Crotch  or  middle 324 

distance 314 

Frustum  of  cone,  Formulas  for 117 

of  pyramid , Formulas  for 118 

Fusion,  Latent  heat  of 18 

Temperature  of 18 

G. 

G2,  Values  of 153 

Galvanometer 271 

Gases,  Weights  of 14 

Gas  lighting,  Electric 269 

Gauge,  Birmingham  wire 249 

B.&S.  wire 248 


INDEX.' 


Page 

Gauge,  sizes  of  wire,  with  equivalent  sectional  areas  248 

Gearing,  Formulas  for 228 

Gears,  Change,  for  screw  cutting 178 

To  calculate  speed  of 142 

Gibs  for  connecting-rods 224 

Gland 228 

Grade  lines 296 

Rate  of 296 

Gravity,  To  find  center  of < 121 

Grindstone,  Speed  of 176 

Gyration,  Square  of  least  radius  of 153 

To  find  radius  of , 131 

To  find  radius  of,  experimentally 133 

H. 

Hangers,  Shaft 202 

Heat 19 

Latent,  of  fusion. 18 

of  liquid 25 

Heating  by  exhaust  steam 173 

of  dynamos 259 

surface,  Square  feet  of,  per  horsepower 168,  169 

surface,  Ratio  of,  to  grate  area 168 

Helix,  Formula  for 116 

To  construct  a 116 

High-speed  engines.  Crank-shaft  for 223 

Horsepower  of  belts 140 

of  boilers 168 

of  electrical  currents 232 

of  engines 185 

of  pumps  . ._ 184 

of  rope  belting 210 

Theoretical 184 

Hydrokinetics 145 

Hydromechanics 144 

Hydrostatics 144 

Hyperbolic  logarithms 32 


I. 

I,  Values  of ' 153 

Incandescent  lamp  data 240 

wires..  Underwriters’  regulations  for 245 

Inch,  Equivalent  decimal  parts  of 91 

Inches  and  parts  thereof  in  decimals  of  one  foot.  ...  91, 92 

Inclined  planes,  Formula  for 138 

Incrustation  in  boilers 164 

Indicated  horsepower  of  engines,  Formula  for 185 

Inertia,  To  find  moment  of 125 

To  find  moment  of,  experimentally.. 133 

To  find  moment  of,  for  various  sections 153 

Inspection  of  boilers 162 


INDEX.  xi 

Page 

Insulation,  Test  of 272 

Interior  wiring 235 

Involution  by  logarithms 44 

Iron  bars,  Weight  of  round  and  square 21 

Irregular  areas 119 

J. 

Joint  coupling,  Universal 195 

Journal  box,  Design  of 195 

K. 

Kerosene  in  boilers 167 

Keys  for  shafting,  Proportions  of 194 

Kilowatt 232 

L. 

Lamps,  Efficiency  of 240 

Incandescent,  data 240 

in  series  (electric  light) • 253 

Lap,  Inside  and  outside 187 

Latent  heat  of  fusion 18 

heat  of  vaporization 18 

Lathe,  Change  gears  of 178 

Compound-geared 182 

Simple-geared 178 

Law,  Ohm’s . 231 

Lead  of  valve 187 

Lead,  Weight  of  sheet 21 

Leakage,  Magnetic 258 

Leclanche  cell 244,269 

Legal  ohm 230 

Length,  Measures  of 5 

Leveling,  Direct 292 

Grade  lines  in 296 

notes,  How  to  check  and  keep 294 

Profiles  in 296 

Levers 136 

Linear  expansion,  Coefficient  of 18 

measure 1 

Line  shafting >y. . . . 193 

Lines  of  force,  Leakage  of 257 

of  force,  Number  of 254 

Lining  for  seats 216 

Liquid,  Heat  of 25 

measures. 4 

Liquids,  Weights  of 13 

Locknuts 192 

Logarithmic  table 50-67 

table,  Use  of 34 

Logarithms 32 

Long-ton  table 3 


INDEX. 


xii 

M • Page 

Machine  design 175 

tools,  Cutting  speeds  for 176 

tools,  Motors  for 263 

Magnetic  meridian 277 

permeability 257 

Manila  rope  belting • 209 

rope  belting,  Weight  of '.  . . . 209 

Mantissa  of  a logarithm. 33 

Materials,  Strength  of - 150 

Mean  effective  pressure 185 

Measure,  Cubic 2 

Dry 3 

Linear 1 

Liquid 4 

Surveyor’s 1 

Surveyor’s  square 2 

Measures  and  weights 1-4 

and  weights,  Metric 5 

of  angles  or  arcs. 2 

of  capacity 6 

of  length 5 

*of  surface  (not  land) 5 

of  volume 5 

of  weight 6 

Mechanical  powers 136 

Mechanics 120,  149 

Mensuration 113 

Meridian,  Magnetic. 277 

True 277 

Metals,  Weights  of 10 

Metric  and  English  measures,  Conversion  table  for.  . 7 

system 5 

Mil .'. 2.35 

Mil,  Circular 235 

Miscellaneous  table 4 

Moment  of  inertia  defined 125 

of  inertia  of  various  sections 153 

of  inertia,  To  find,  experimentally 133 

of  resistance  defined 134 

of  resistance  of  various  sections 133 

Moments,  Bending 152 

Mo  to  r ci  rcuits 253 

efficiency,  Approximate. . .. 253 

Motors,  Application  of  electric 261 

for  machine  tools 263 

Output  and  efficiency  of 260 

Polarity  of 234 

Underwriters’  rules  for 247 

Multiple  arc,  Lamps  in -. -.  . . . 253 

Multiplication  by  logarithms 41 


INDEX. 


xiii 


N.  Page 

Needle,  Declination  of 277 

Neutral  axis 135 

Numbers,  Prime 79 

Nuts,  Proportions  of 22 


o. 

Oblique  fixed  pulley.  Formula  for. . 

Ohm,  Legal 

Ohm’s  law 

Oil  cup 

Oscillation,  To  find  center  of 

To  find  radius  of,  experimentally. 
Output  of  motors 


138 

230 

231 
198 
127 
133 
260 


P. 


Packing  rings 224 

Paper,  To  make  blueprint. 175 

Parallel,  Lamps  in 253 

Parallelogram 114 

of  forces,  Explanation  of 137 

Passage,  Steam 217 

Pedestals,  Design  and  proportions  of 195-202 

Percussion,  To  find  center  of 130 

Permeability,  Magnetic 257 

Perpetual  calendar 327 

Pipe  flanges 215 

Weight  of  cast-iron.  . .• 23 

Pipes  and  cylinders,  Strength  of 157 

Flow  of  water  in 147 

Sizes  of  wrought-iron 24 

Piston  clearance 216 

Pistons 223 

Pitch,  Formula  for  circular 228 

Table  of  circular 230 

Formula  for  diametral 228 

Table  of  diametral 230 

of  bolts  in  cylinder  heads 217 

of  bolts  in  steam-chest  covers 217 

Polarity  of  a dynamo,  To  determine 233 

of  an  electromagnet,  To  determine 233 

Polishing  wheels,  Speed  of 176 

Polygon  of  forces,  Explanation  of 137 

Polygons,  Regular 119 

Port,  Exhaust 217 

Steam 217 

Power  transmitted  by  leather  belting 141 

transmitted  by  rope  belting 209 

Powers,  Mechanical 136 

roots,  and  reciprocals,  Table  of 110 

Pressure,  Mean  effective 185 


XIV 


INDEX. 


Page 

Pressures,  Absolute 26 

Prime  factors,  Table  of 80 

numbers 79 

Priming  of  boilers 168 

Prismoidal  formula 306 

Profiles  in  leveling 296 

Projectiles 120 

Prony  brake 260 

Properties  of  aluminum  and  copper 250 

of  saturated  steam 29 

Proportions  of  belt  pulleys 204-209 

of  flange  couplings 194 

of  journal  boxes 195-202 

of  keys 193 

of  rope-pulley  rims 211 

of  shaft  hangers 202 

Pulleys,  Belt 204 

Differential 138 

Double  movable 137 

Formula  for  compound 138 

Proportions  of 204-209 

Quadruple  movable 138 

Rope 211 

Single  fixed 137 

Single  movable 137 

Speed  of 142 

Pumps,  Discharge  of 143 

Duty  of 144 

Horsepower  of » 143 

Pyramid  Formulas  for ' 118 

Formulas  for  frustum  of 118 

Q. 

Quadruple  movable  pulley,  Formula  for 138 

R. 

R,  Values  oi 153 

Radiating  surface  in  exhaust-steam  heating 174 

Radii  and  deflections.  Table  of 298-300 

Radius  of  gyration,  To  find.  131 

of  gyration,  To  find,  experimentally 133 

of  gyration,  To  find,  for  various  sections 153 

of  oscillation,  To  find,  experimentally 133 

Rate  of  transmission  of  electricity 231 

Reciprocal  of  a number 108 

Rectangle,  Formula  for 114 

Regular  polygons. 119 

Resistance,  Electrical 231 

Moment  of 134 

Moment  of,  for  various  sections 153 

of  copper  wire 251-253 


INDEX. 


xv 


Page 


Resistance  of  derived  circuit 234 

Resultant  of  forces 137 

Retaining  walls 300 

walls,  Resistance  of,  to  overturning 303 

Reversal  of  field 258 

Ribs  for  piston 224 

for  steam-chest  cover 220 

Ring,  Formula  for  circular 117 

Formula  for 116 

Root,  Cube 105 

Square 103 

Roots,  Method  of  extracting 103 

Table  of 110 

Rope  belting 209 

belting,  Pulleys  for 211 

Weight  of  manila 209 

Ropes  and  chains,  Strength  of 157 

Wire 212 

wire,  Pulleys  for 213 


s. 


Safety,  Table  of  factors  of 

valves 

Saturated  steam,  Properties  of 

'Saturation  curves  (electrical) 

Scale  in  boilers,  Prevention  of 

Screw  cutting,  Change  gears  for 

Formulas  for 

threads,  Proportion  of 

Seats,  Lining  for 

Sector,  Formula  for 

Segment,  Formula  for 

Series.  Lamps  in 

Shaft  couplings 

hangers 

Shafting,  Formulas  for 

Line 

Shafts,  Crank 

Shearing  strengths,  Table  of 

Sheaves  for  rope  gearing 

Sheet  lead,  Weight  of 

Shunt  circuit 

Simple-geared  lathe,  Screw  cutting  with 

Single  fixed  pulley,  Formula  for 

movable  pulley,  Formula  for 

Size  of  copper  wire  for  circuits 

Slide  valve 

valve  diagram 

Slope  sheet  of  boiler,  To  develop 

Soldering,  Fluxes  for 

Solders 


151 

173 

29 

256 

164 

178, 

139' 

22 

216 

115 

115 

253 

194 

202 

157 

193 

223 

151 

213 

21 

234 

178 

137 

137 

235-252 

187 

188 
160 

24 

20 


XVI 


INDEX. 


Sparking  at  commutator 

Specific  gravity,  Table  of 

heat,  Table  of 

volumes 

Sp>eed,  Cutting ' ... . 

of  emery  wheels 

of  gears,  To  calculate 

of  grindstones 

of  polishing  wheels 

of  pulleys,  To  calculate 

of  twist  drills 

Sphere,  Formula  for 

Spiral,  Length  of 

Square  measure 

root 

Squares  and  cubes 

Standard  pipe  flanges 

Steam  chest 

chest  bolts 

chest  covers 

Heating  by  exhaust 

port  area 

Properties  of  saturated 

- tables 

Velocity  of,  through  ports 

Steel,  Tempering  of 

Stone,  Weight  of 

Storage  batteries 

Strands  in  wire  rope 

Strap,  Eccentric  and 

end  of  connecting-rod 

Strength  of  materials 

Stroke  of  engine 

Stuffingbox 

Surcharged  walls,  Pressure  on. . . . 
Surface  expansion,  Coefficients  of 

Measures  of  (not  land) 

Surveying 

with  compass 

with  transit 

Surveyor’s  measure. 

square  measure 

Switch 

Point  or  split 

Stub 

stub,  To  lay  out 

Systems,  Annunciator 


Page 

259 

10 

18 

25 

176 

176 

142 

176 

176 
142 

177 

117 

118 
2 

103 

106 

215 

219 

220 
221 

173, 174 
219 
29 
25 
219 
6 

10 

263 

212 

226 

224 

150 

216 
228 
305 

19 

5 

276-326 

276 

280 

1 


315 

315 

315 

320 

243 


T. 


Table,  Long- ton 3 

Miscellaneous 4 


INDEX.  xvii 

Page 

Table  of  chemical  elements 16 

of  powers,  roots,  and  reciprocals 110 

Tables,  Steam 25 

Wire 247-251 

Teeth  of  wheels 228 

Temperature  of  fusion 18 

of  vaporization 18 

Tempering  steel 6 

Tensile  strength  of  materials 151 

Tension  of  rope  belting,  Formula  for 210 

Testing  of  cables  (electrical) 271 

Threads,  Cutting  screw 178 

Proportions  of  screw 22 

Three-wire  system,  Edison 253 

Ton,  Long 3 

Tools,  Cutting  speeds  for  machine 176 

cutting,  Motors  for 263 

Torque 261 

Tracks,  Cross-over 325 

Trackwork 309 

Transit  notes,  How  to  keep 291 

surveying 279 

Trapezium  and  trapezoid,  Formula  for 115 

• Triangles,  Formulas  for 114 

Triangulation 283 

Trigonometric  functions,  Directions  for  use  of  table  of  68 

functions,  Table  of 74-78 

Troy  weight 3 

Tunnel  sections 306 

Turnouts 312 

Type  metals 15 


U. 

Ultimate  strength  of  materials 151 

Underwriters’  line  wire.  . . . 247 

rules  for  incandescent  wire 245 

Units,  Electrical 230 

Universal  joint  coupling '. 195 

Useful  tables 1-92 

V. 

Valve  diagram 188 

Valves,  Safety 173 

Slide 187 

Vaporization,  Latent  heat  of 18,  25 

Temperature  of . . . .* 18 

Vapors,  Weights  of 14 

Velocity  of  steam  through  ports 219 

Vernier 279 

Volt 230 

Volume,  Measures  of 5 

Volumes,  Specific 25 


INDEX. 


xviii 


w. 


Page 


Water,  Testing  of  feed 

Flow  of,  in  pipes 

Watt,  The  unit .' 

Wedge,  Formula  for 

Weight,  Avoirdupois 

Measures  of 

of  bar  iron,  round  and  square 

of  copper  wire 

of  manila  rope 

of  sheet  lead 

of  various  substances 

Troy 

Weights  and  measures 

and  measures,  Metric  system  of 

Welding  fluxes  . 

Wheatstone  bridge 

Wheel  and  axle 

Wheels,  Speed  of  emery 

Speed  of  polishing 

Wheel  work,  Formulas  for 

Width  of  belts.  Formulas  for 

Wire,  copper,  Sizes  for  circuit 

copper,  Weight  of 

gauges,  Sizes  of  B.  & S.  and  Birmingham 

rope,  Steel.  

rope,  Strands  in 

tables 

Underwriters’  line 

Wires,  Equivalent  areas  of,  B.  & S.  gauge. . 

Wiring,  Bell.- 

Interior 

Work,  Definition  of 

Wristpin  brasses 

Wrought-iron  pipe,  Sizes  of 


165 

147 

231 

117 

3 

6 

21 

238 

209 

21 

10 

3 

1-4 


5-6 

24 

270 

136 

176 

176 

136 

140 

.235-240 

238 

249 

212 

212 

.247-251 

247 

248 
241 
235 
139 
224 

24 


8 88*.  8 8 


Mechanics’ 
Pocket  Memoranda 


USEFUL  TABLES. 


WEIGHTS  AND  MEASURES. 


12 

inches  (in.)  . 

LINEAR  MEASURE. 

— 1 foot  

3 

feet  

= 1 yard 

5.5  yards  

= 1 rod  

40 

rods 

= 1 furlong 

8 

furlongs 

= 1 mile  ... 

in. 

ft.  yd.  rd.  fi 

36 

= 3 = 1 

198 

= 16.5  - 5.5  = 1 

7,920 

II 

05 

o 

11 

to 

8 

II 

o 

II 

63,360 

= 5,280  = 1,760  = 320  = 

...ft. 

.yd. 

..rd. 

.fur. 

.mi. 


SURVEYOR’S  MEASURE. 


inches  = 1 link  

links = lrod 

rods  ) 

links  > = 1 chain  

feet  j 

chains = 1 mile. 

1 mi.  = 80  ch.  = 320  rd.  = 8,000  li.  = 63,360  in. 


...li. 

.rd. 

.ch, 

.mi. 


1 


2 


USEFUL  TABLES. 


SQUARE  MEASURE. 


144  square  inches  (sq.  in.). 

9 square  feet 

30£  square  yards 

160  square  rods  

640  acres 


= 1 square  foot sq.  ft. 

= 1 square  yard sq.  yd. 

= 1 square  rod sq.  rd. 

= 1 acre  A. 

= 1 square  mile sq.  mi. 


sq.  mi.  A.  sq.  rd.  sq.  yd.  sq.  ft.  sq.  in. 

1 = m = 102,400  = 3,097,600  = 27,878,400  = 4,014,489,600 


SURVEYOR'S  SQUARE  MEASURE. 

625  square  links  (sq.  li.)  = 1 square  rod sq.  rd. 

16  square  rods = 1 square  chain  sq.  ch. 

10  square  chains = 1 acre A. 

640  acres = 1 square  mile sq.  mi. 

36  square  miles  (6  mi.  square)  = 1 township Tp. 

1 sq.  mi.  = 640  A.  = 6,400  sq.  ch.  = 102,400  sq.  rd. 

= 64,000,000  sq.  li. 

The  acre  contains  4,840  sq.  yd.,  or  43,560  sq.  ft.,  and  is 
equal  to  the  area  of  a square  measuring  208.71  ft.  on  a side. 


CUBIC  MEASURE. 

1,728  cubic  inches  (cu.  in.) = 1 cubic  foot  .. 

27  cubic  feet  = 1.  cubic  yard... 

128  cubic  feet  = 1 cord 

24£  cubic  feet = 1 perch  

1 cu.  yd.  = 27  cu.  ft.  = 46,656  cu.  in 


MEASURE  OF  ANGLES  OR  ARCS. 

60  seconds  (") * = 1 minute  ' 

60  minutes  — 1 degree  ° 

90  degrees = 1 rt.  angle  or  quadrant  □ 

360  degrees ..  = 1 circle cir. 

1 cir.  = 360°  = 21,600'  = 1,296,000" 


..cu.  ft. 
.cu.  yd. 

cd. 

P. 


WEIGHTS  AND  MEASURES. 


3 


AVOIRDUPOIS  WEIGHT. 

437.5  grains  (gr.) = 1 ounce oz. 

16  ounces  = 1 pound lb. 

100  pounds = 1 hundredweight  cwt. 

20  cwt.,  or  2,0001b = 1 ton T. 

IT.  = 20  cwt.  = 2,0001b.  = 32,000  oz.  = 14,000,000 gr. 
The  avoirdupois  pound  contains  7,000  grains. 


16  ounces  

LONG  TON  TABLE. 

....  — 1 pound 

lb. 

112  pounds 

— 1 hundredweight  .. 

....cwt. 

20  cwt.,  or  2,240  lb. 

— 1 ton 

T. 

24  grains  (gr.)  

TROY  WEIGHT. 

....pwt. 

20  pennyweights 

s=3  1 ounce 

oz* 

12  ounces 

— 1 pound 

lb* 

lib.  = 

12  oz.  = 240  pwt.  = 5,760  gr. 

2pints(pt.)  

DRY  MEASURE. 

qt. 

8 quarts  

..: = 1 peck  

pk. 

4 pecks 

= 1 bushel  

bu. 

lbu.  = 4pk.  = 32  qt.=  64  pt. 

The  U.  S.  struck  bushel  contains  2,150.42  cu.  in.  = 1.2444 
cu.  ft.  • By  law,  its  dimensions  are  those  of  a cylinder  18£  in. 
in  diameter  and  8 in.  deep.  The  heaped  bushel  is  equal  to 
1£  struck  bushels,  the  cone  being  6 in.  high.  The  dry  gallon 
contains  268.8  cu.  in.,  being  £ of  a struck  bushel. 

For  approximations,  the  bushel  may  be  taken  at  1£  cu.  ft.; 
or  a cubic  foot  may  be  considered  £ of  a bushel. 

The  British  bushel  contains  2,218.19  cu,  in.  = 1.2837  cu.  ft. 
= 1.032  U.  S.  bushels. 


4 


USEFUL  TABLES. 


LIQUID  MEASURE. 


4 gills  (gi.)  

=*1  pint  

Pt. 

2 pints  

= 1 quart  

-v qt. 

4 quarts 

= 1 gallon  

gal. 

31£  gallons 

= 1 barrel  

bbl. 

2 barrels,  or  63  gallons  .... 

s=  1 hogshead 

hhd. 

1 hhd.  = 2 bbl.  = 63  gal.  = 252  qt.  = 504  pt.  = 2,016  gi. 


The  U.  S.  gallon  contains  231  cu.  in.  = .134  cu.  ft.,  nearly; 
or  1 cu.  ft.  contains  7.481  gal.  The  following  cylinders  con- 
tain the  given  measures  very  closely: 


Diam. 

Height. 

Diam. 

Height. 

Gill 

If  in. 

3 in. 

Gallon  

. 7 in. 

6 in. 

Pint  

....  3£in. 

3 in. 

8 gallons 

Min. 

12  in. 

Quart  ... 

....  3*  in. 

6 in. 

10  gallons 

14  in. 

15  in. 

When  water  is  at  its  maximum  density,  1 cu.  ft.  weighs 
62.425  lb.  and  1 gallon  weighs  8.345  lb. 

For  approximations,  1 cu.  ft.  of  water  is  considered  equal 
to  7i  gal.,  and  1 gal.  as  weighing  8|  lb. 

The  British  imperial  gallon,  both  liquid  and  dry,  con- 
tains 277.274  cu.  in.  = .16046  cu.  ft.,  and  is  equivalent  to  the 
volume  of  10  lb.  of  pure  water  at  62°  F.  To  reduce  British  to 
U.  S.  liquid  gallons,  multiply  by  1.2.  Conversely,  to  convert 
U.  S.  into  British  liquid  gallons,  divide  by  1.2;  or,  increase 
the  number  of  gallons 


MISCELLANEOUS  TABLE. 


12  articles  = 

1 dozen. 

20  quires  = 1 ream. 

12  dozen  = 

1 gross. 

1 league  = 3 miles. 

12  gross  = 

1 great  gross. 

1 fathom  = 6 feet. 

2 articles  = 

1 pair. 

1 hand  = 4 inches. 

20  articles  = 

1 score. 

1 palm  = 3 inches. 

24  sheets  = 

1 quire. 

1 span  = 9 inches. 

1 sea  mile  (U.  S.)  = 6,080  ft.  = 1£  statute  miles  (roughly). 
1 meter  = 3 feet  3|  inches  (nearly). 


THE  METRIC  SYSTEM. 


5 


THE  METRIC  SYSTEM. 

The  metric  system  is  based  on  the  meter,  which,  according 
to  the  U.  S.  Coast  and  Geodetic  Survey  Report  of  1884,  is  equal 
to  39.370432  inches.  The  value  commonly  used  is  39.37  inches, 
and  is  authorized  by  the  U.  S.  government.  The  meter  is 
defined  as  one  ten-millionth  the  distance  from  the  pole  to  the 
equator,  measured  on  a meridian  passing  near  Paris. 

There  are  three  principal  units— the  meter,  the  liter  (pro- 
nounced lee-ter),  and  the  gram,  the  units  of  length,  capacity, 
and  weight,  respectively.  Multiples  of  these  units  are  obtained 
by  prefixing  to  the  names  of  the  principal  units  the  Greek 
words  deca  (10),  hecto  (100),  and  kilo  (1,000);  the  submulti- 
ples, or  divisions,  are  obtained  by  prefixing  the  Latin  words 
deci  (TV),  centres),  and  milli  (nrW)-  These  prefixes  form 
the  key  to  the  entire  system.  In  the  following  tables,  the 
abbreviations  of  the  principal  units  of  these  submultiples  begin 
with  a small  letter,  while  those  of  the  multiples  begin  with  a 
capital  letter;  they  should  always  be  written  as  here  printed. 


MEASURES 

OF  LENGTH. 

10  millimeters  (mm.)  

— 1 centimeter 

cm. 

10  centimeters 

= 1 decimeter  

dm. 

10  decimeters  

= 1 meter  

m. 

10  meters  

— 1 decameter 

Dm. 

10  decameters 

= 1 hectometer 

Hm. 

10  hectometers 

= 1 kilometer  

Km. 

MEASURES  OF  SURFACE  (NOT  LAND). 

100  square  millimeters  (mm2.)  = 1 square  centimeter cm2. 

100  square  centimeters  = 1 square  decimeter  dm2. 

100  square  decimeters = 1 square  meter  m2. 


MEASURES  OF  VOLUME. 

1,000  cubic  millimeters  (mm3.)  = 1 cubic  centimeter cm3. 

1,000  cubic  centimeters  = 1 cubic  decimeter dm3. 

1,000  cubic  decimeters »=  1 cubic  meter m3. 


6 


USEFUL  TAELES. 


M EASURES 

OF  CAPACITY. 

10  milliliters  (ml.) 

= 1 centiliter. 

cl. 

10  centiliters  

= 1 deciliter 

dl 

10  deciliters 

= 1 liter 

1. 

10  liters  

— 1 decaliter  

Dl. 

10  decaliters 

. — 1 hectoliters  

HI. 

10  hectoliters  

= 1 kiloliters  

Kl. 

Note. — The  liter  is  equal  lo  the  volume  that  is  occupied 

by  1 cubic  decimeter. 

MEASURES 

OF  WEIGHT. 

10  milligrams  (mg.) 

= 1 centigram 

eg. 

10  centigrams 

. = 1 decigram  

dg. 

10  decigrams  

= 1 gram  

g. 

10  grams 

,=  1 decagram 

Dg. 

10  decagrams  

= 1 hectogram 

Hg. 

10  hectograms 

= 1 kilogram 

Kg. 

1,000  kilograms  

= 1 ton 

T. 

Note.— The  gram  is  the  weight  of  1 cubic  centimeter  of 
pure  distilled  water  at  a temperature  of  39.2°  F. ; the  kilogram 
is  the  weight  of  1 liter  of  water;  the  ton  is  the  weight  of 
1 cubic  meter  of  water. 


TEMPERING  OF  STEEL. 

The  following  colors  may  be  made  use  of  in  tempering 


steel-cutting  tools: 


Corresponding 
Temperature  F. 


Lancets 

Razors  

All  kinds  of  wood-cutting 

tools 

Screw  taps  

Chipping  chisels,  hatchets, 

and  saws  

All  kinds  of  percussive  tools  y 

Springs  j 


Pale  yellow 430° 

Straw  yellow 450° 

Darker  straw  yellow  470° 

Yellow  490° 

Brown  yellow 500° 

Brown  (slightly  tinged 

purple) 520° 

Light  purple  530° 

Clear  black 570° 

Dark  blue 600° 


THE  METRIC  SYSTEM. 


7 


1,828.8 

121.92 

21.336 

.3048 

.2438 

1,972.6046 


CONVERSION  TABLES. 

By  means  of  the  tables  on  pages  8 and  9,  metric  measures 
can  be  converted  into  English,  and  vice  versa , by  simple  addi- 
tion. All  the  figures  of  the  values  given  are  not  required, 
four  or  five  digits  being  all  that  are  commonly  used;  it  is 
only  in  very  exact  calculations  that  all  the  digits 
are  necessary.  Using  table,  proceed  as  follows: 

Change  6,471.8  feet  into  meters.  Any  number,  as 
6,471.8,  may  be  regarded  as  6,000  + 400  + 70  + 1 
+ .8;  also,  6,000  = 1,000  X 6;  400  = 100  X 4,  etc. 

Hence,  looking  in  the  left-hand  column  of  the 
upper  table,  page  8,  for  figure  6 (the  first  figure  of 
the  given  number),  we  find  opposite  it  in  the  third 
column,  which  is  headed  “Feet  to  Meters,”  the  number 
1.8287838.  Now,  using  but  five  digits  and  increasing  the  fifth 
digit  by  1 (since  the  next  is  greater  than  5) , we  get  1.8288.  In 
other  words,  6 feet  = 1.8288  meters;  hence,  6,000  feet  = 1,000 
X 1.8288  = 1,828.8,  simply  moving  the  decimal  point  three 
places  to  the  right.  Likewise,  400  feet  = 121.92  meters;  70  feet 
= 21.336  meters;  1 foot  = .3048  meter,  and  .8  foot  = .24C"  meter. 
Adding  as  shown  above,  we  get  1,972.6046  meters. 

Again,  convert  19.635  kilos  into  pounds.  The 
work  should  be  perfectly  clear  from  the  explana- 
tion given  above.  The  result  is  43.2875  pounds. 

The  only  difficulty  m applying  these  tables  lies 
in  locating  the  decimal  point;  it  may  always  be 
found  thus:  If  the  figure  considered  lies  to  the  left 
of  the  decimal  point,  count  each  figure  in  order, 
beginning  with  units  (but  calling  unit’s  place  zero),  until  the 
desired  figure  is  reached,  then  move  the  decimal  point  to 
the  right  as  many  places  as  the  figure  being  considered  is  to 
the  left  of  the  unit  figure.  Thus,  in  the  'first  case  above, 
6 lies  three  places  to  the  left  of  1,  which  is  in  unit’s  place; 
hence,  the  decimal  point  is  moved  three  places  to  the  right. 
By  exchanging  the  words  “right ” and  “ left,”  the  statement 
will  also  apply  to  decimals.  Thus,  in  the  second  case  above, 
the  5 lies  three  places  to  the  right  of  unit’s  place;  hence,  the 
decimal  point  in  the  number  taken  from  the  table  is  moved 
three  places  to  the  left 


22.046 

19.8416 

1.3228 

.0661 

.0110 

43.2875 


8 


USEFUL  TABLES. 


Conversion  Table— English  Measures  Into  Metric. 


English. 

Metric. 

Metric. 

Metric. 

Metric. 

Inches  to 
Meters. 

Feet  to 
Meters. 

Pounds  to 
Kilos. 

Gallons  to 
Liters. 

1 

.0253998 

.3047973 

.4535925 

3.7853122 

2 

.0507996 

.6095946 

.9071850 

7.5706244 

3 

.0761993 

.9143919 

1.3607775 

11.3559366 

4 

.1015991 

1.2191892 

1.8143700 

15.1412488 

5 

.1269989 

1.5239865 

2.2679625 

18.9265610 

6 

.1523987 

1.8287838 

2.7215550 

22.7118732 

7 

.1777984 

2.1335811 

3.1751475 

26.4971854 

8 

.2031982 

2.4383784 

3.6287400 

30.2824976 

9 

.2285980 

2.7431757 

4.0823325 

34.0678098 

10 

.2539978 

3.0479730 

4.5359250 

37.8531220 

[ 

Conversion  Table— English  Measures  Into  Metric. 


English. 

Metric. 

Metric. 

Metric. 

Metric. 

Square 

Inches 

to 

Square 

Meters. 

Square 

Feet 

to 

Square 

Meters. 

Cubic 

Feet 

to 

Cubic 

Meters. 

Pounds  per 
Square  Inch 
to  Kilo  per 
Square 
Metgr. 

1 

.000645150 

.092901394 

.028316094 

703.08241 

2 

.001290300 

.185802788 

.056632188 

1,406.16482 

3 

.001935450 

.278704182 

.084948282 

2,109.24723 

4 

.002580600 

.371605576 

.113264376 

2,812.32964 

5 

.003225750 

.464506970 

.141580470 

3,515.41205 

6 

.003870900 

.557408364 

.169896564 

4,218.49446 

7 

.004516050 

.650309758 

.198212658 

4,921.57687 

8 

.005161200 

.743211152 

.226528752 

5,624.65928 

9 

.005406350 

.836112546 

.254844846 

6,327.74169 

10 

.006451500 

.929013940 

.283160940 

7,030.82410 

THE  METRIC  SYSTEM. 


9 


Conversion  Table— Metric  Measures  Into  English. 


Metric. 

English. 

English. 

English. 

English. 

Meters  to 
Inches. 

Meters  to 
Feet. 

Kilos  to 
Pounds. 

Liters  to 
Gallons. 

1 

39.370432 

3.2808693 

2.2046223 

.2641790 

. 2 

78.740864 

6.5617386 

4.4092447 

.5283580 

3 

118.111296 

9.8426079 

6.6138670 

.7925371 

4 

157.481728 

13.1234772 

8.8184894 

1.0567161 

5 

196.852160 

16.4043465 

11.0231117 

1.3208951 

6 

236.222592 

19.6852158 

13.2277340 

1.5850741 

7 

275.593024 

22.9660851 

15.4323564 

1.8492531 

8 

314.963456 

26.2469544 

17.6369787 

2.1134322 

9 

354.333888 

29.5278237 

19.8416011 

2.3776112 

10 

393.704320 

32.8086930 

22.0462234 

2.6417902 

Conversion  Table— Metric  Measures  Into  English. 


Metric. 

English. 

English. 

English. 

English. 

Square 

Meters 

to 

Square 

Inches. 

Square 

Meters 

to 

Square 
- Feet. 

Cubic 

Meters 

to 

Cubic 

Feet. 

Kilos  per 
Square 
Meter  to 
Pounds  per 
Square 
Inch. 

1 

1,550.03092 

10.7641034 

35.3156163 

.001422310 

2 

3,100.06184 

21.5282068 

70.6312326 

.002844620 

3 

4,650.09276 

32.2923102 

105.9468489 

.004266930 

4 

6,200.12368 

43.0564136 

141.2624652 

.005689240 

5 

7,750.15460 

53.8205170 

176.5780815 

.007111550 

6 

9,300.18552 

64.5846204 

211.8936978 

.008533860 

7 

10,850.21644 

75.3487238 

247.2093141 

.009956170 

8 

12,400.24736 

86.1128272 

282.5249304 

.011378480 

9 

13,950.27828 

96.8769306 

317.8405467 

.012800790 

10 

15,500.30920 

107.6410340 

353.1561630 

.014223100 

i 

10 


USEFUL  TABLES. 


SPECIFIC  GRAVITY. 

The  specific  gravity  of  a body  is  the  ratio  between  its 
weight  and  the  weight  of  a like  volume  of  distilled  water  at 
a temperature  of  39.2°  F.  For  gases,  air  is  taken  as  the  unit. 
One  cubic  foot  of  water  at  39.2°  F.  weighs  62.425  pounds. 


Name  of  Substance. 

Specific 

Gravity. 

Weight 
per  Cu.  In. 
Pounds. 

Metals. 

Platinum,  rolled 

22.009 

.819 

Platinum,  wire  

21.042 

.760 

Platinum,  hammered 

20.337 

.735 

Gold,  hammered 

19.361 

.699 

Gold,  pure  cast  

19.258 

.696 

Gold,  22  carats  fine 

17.486 

.632 

Mercury,  solid  at  — 40°  F 

15.632 

.565 

Mercury,  at  + 32°  F. 

13.619 

.492 

Mercury,  at  60°  F. 

13.580 

.491 

Mercury,  at  212°  F 

13.375 

.483 

Lead,  pure 

11.330 

.409 

Lead,  hammered  

11.388 

.411 

Silver,  hammered 

10.511 

.380 

Silver,  pure 

10.474 

.378 

Bismuth 

9.746 

.352 

Copper,  wire  and  rolled 

8.878 

.321 

Copper,  pure  

8.788 

.317 

Bronze,  gun  metal 

8.500 

.307 

Brass,  common  

8.500 

.307 

Steel,  cast  steel 

7.919 

.286 

Steel,  common  soft 

7.833 

.283 

Steel,  hardened  and  tempered 

7.818 

.282 

Iron,  pure 

7.768 

.281 

Iron,  wrought  and  rolled 

7.780 

.281 

Iron,  hammered 

7.789 

.281 

Iron,  cast  

7.207 

.260 

Tin,  from  Bohmen  

7.312 

.264 

Tin,  English 

7.201 

.263 

Zinc,  rolled 

7.101 

.260 

Antimony 

6.712 

.242 

Aluminum  

2.660 

.096 

Stones  and  Earths. 

Emery 

4.000 

.145 

Limestone 

2.700 

.098 

Asbestos,  starry 

3.073 

.111 

SPECIFIC  GRAVITY. 


11 


Table— ( Continued)  * 


Name  of  Substance. 


Specific 

Gravity. 


Weight 
per  Cu.  In. 
Pounds. 


Glass,  flint 

Glass,  white 

Glass,  bottle 

Glass,  green 

Marble,  Parian  ... 
Marble,  African... 
Marble,  Egyptian 
Mica 


3.500 

2.900 

2.732 

2.642 

2.838 


.1260 

.1050 

.0987 

.0954 

.1025 


2.708 

2.668 

2.800 


.0978 

.0964 

.1012 


Chalk 


2.784 


.1006 


Coral,  red 

Granite,  Susquehanna. 

Granite,  Quincy 

Granite,  Patapsco 

Granite,  Scotch 

I Marble,  white  Italian 

Marble,  common  

| Talc,  black  

Quartz 

Slate  

Pearl,  oriental 

Shale  

Flint,  white 

Flint,  black 

Stone,  common  

Stone,  Bristol  

Stone,  mill  

Stone,  paving 

Gypsum,  opaque 

Grindstone  

Salt,  common 

Saltpeter  

Sulphur,  native 

Common  soil  

I Rotten  stone  

Clay 

Brick  

Niter  

Plaster  Paris 

1 Ivor^  

J Sand  

Phosphorus 

| Borax 

| Coal,  anthracite 


2.700 

2.704 

2.652 

2.640 
2.625 
2.708 
2.686 

2.900 
2.660 
2.800 
2.650 
2.600 
2.594 
2.582 
2.520 
2.510 
2.484 
2.416 
2.168 
2.143 
2.130 
2.090 
2.033 
1.984 
1.981 

1.900 
2.000 
1.900 
1.872 
2.473 
1.822 
2.650 
1.770 
1.714 

1.640 
1.436 


.0975 

.0977 

.0958 

.0954 

.0948 

.0978 

.0970 

.0105 

.0961 

.1012 

.0957 

.0939 

.0937 

.0933 

.0910 

.0907 

.0897 

.0873 

.0783 

.0774 

.0769 

.0755 

.0734 

.0717 

.0716 

.0686 

.0723 

.0686 

.0676 

.0893 

.0659 

.0957 

.0639 

.0619 

.0592 

.0519 


12 


USEFUL  TABLES. 


Table— ( Continued). 


Name  of  Substance. 

Specific 

Gravity. 

Weight 
per  Cu.  In. 
Pounds. 

Coal,  Maryland 

1.355 

.0490 

Coal,  Scotch 

1.300 

.0470 

Coal,  Newcastle 

1.270 

.0459 

Coal,  bituminous  

1.350 

.0488 

Earth,  loose 

1.360 

.0491 

Lime,  quick 

1.500 

.0542 

Charcoal  .* 

.441 

.0159 

Woods  (Dry). 

Alder  

.800 

.0289 

Apple  tree 

.793 

.0287 

Ash,  the  trunk  

.845 

.0305 

Bay  tree 

.822 

.0297 

Beech 

.852 

.0308 

Box,  French 

.960 

.0347 

Box,  Dutch  

1.328 

.0480 

Box,  Brazilian  red 

1.031 

.0372 

Cedar,  wild  

.596 

.0215 

Cedar,  Palestine 

.613 

.0221 

Cedar,  American  

.561 

.0203 

Cherry  tree  

.672 

.0243 

Cork 

.250 

.0090 

Ebony,  American 

1.220 

.0441 

Elder  tree *. 

.695 

.0251 

Elm 

.560 

.0202 

Filbert  tree  

.600 

.0217 

Fir,  male  

.550 

.0199 

Fir,  female  

.498 

.0180 

Hazel  

.600 

.0217 

Lemon  tree  

.703 

.0254 

Lignum-vitse  

1.330 

.0481 

Linden  tree 

.604 

.0218 

Logwood  

Mahogany,  Honduras 

.913 

.0330 

.560 

.0202 

Maple 

.790 

.0285 

Mulberry  

.897 

.0324 

Oak 

.950 

.0343 

Orange  tree 

.705 

.0255  . 

Pear  tree  

.661 

.0239 

Poplar 

.383 

.0138 

Poplar,  white  Spanish 

.529 

.0191 

Sassafras 

.482 

.0174 

Spruce 

.500 

.0181 

Spruce,  old  

.460 

.0166 

SPECIFIC  GRAVITY. 


13 


Table— ( Continued). 


Name  of  Substance. 


Specific 

Gravity. 


Pine,  southern 
Pine,  white  .... 
Walnut  


.720 

.400 

.610 


Liquids. 

Acid,  acetic ....... 

Acid,  nitric  .7 

Acid,  sulphuric 

Acid,  muriatic - : 

Acid,  phosphoric  

Alcohol,  commercial  

Alcohol,  pure ..... 

Beer,  lager 

Champagne  

Cider  , 

Ether,  sulphuric 

Egg  

Honey 

Human  blood - 

Milk 

Oil,  linseed  

Oil,  olive  

Oil,  turpentine  

Oil,  whale 

Proof  spirit  

Vinegar  

Water,  distilled  (62.425  lb.  per  cu.  ft.) 

Water,  sea 

Wine  


1.062 

1.217 

1.841 

1.200 

1.558 

.S33 

.792 

1.034 

.997 

1.018 

.739 

1.090 

1.450 

1.054 

1.032 


.915 

.870 

.932 

.925 

1.080 

1.000 

1.030 

.992 


Miscellaneous. 

Beeswax  

Butter 

India  rubber 

Fat  

Gunpowder,  loose 

Gunpowder,  shaken 

Gum  arabic  

Lard 

Spermaceti  

Sugar  

Tallow,  sheep 

Tallow,  calf 

Tallow,  ox 

Atmospheric  air 


.965 

.942 

.933 

.923 

.900 

1.000 

1.452 

.947 

.943 

1.605 

.924 

.934 

.923 

.0012 


Weight 
per  Cu.  In. 
i Pounds. 


.0260 

.0144 

.0220 


.0384 

.0440 

.0665 

.0434 

.0563 

.0301 

.0286 

.0374 

.0360 

.0368 

.0267 

.0394 

.0524 

.0381 

.0373 

.0340 

.0331 

.0314 

.0337 

.0334 

.0390 

.0361 

.0372 

.0358 


.0349 

.0340 

.0337 

.0333 

.0325 

.0361 

.0525 

.0342 

.0341 

.0580 

.0334 

.0337 

.0333 


14 


USEFUL  TABLES. 


Table — ( Continued) . 


Name  of  Substance. 

Specific 

Gravity. 

Weight 
per  Cu.  Ft. 
Grains. 

Gases  and  Vapors. 

At  32°  and  a tension  of  1 atmosphere. 

Atmospheric  air 

1.0000 

565.11 

Ammonia  gas 

.5894 

333.1  ' 

Carbonic  acid 

1.5201 

859.0 

Carbonic  oxide  

.9673 

546.6 

Light  carbureted  hydrogen 

.5527 

312.3 

Chlorine 

2.4502 

1,384.6 

Olefiant  gas 

.9672 

546.6 

Hydrogen  

.0692 

39.1 

Oxygen  

1.1056 

624.8 

Sulphureted  hydrogen 

1.1747 

663.8 

Nitrogen 

.9713 

548.9 

Vapor  of  alcohol 

1.5890 

898.0 

Vapor  of  turpentine  spirits 

4.6978 

2,654.8 

Vapor  of  water 

.6219 

351.4 

Smoke  of  bituminous  coal 

.1020 

57.6 

Smoke  of  wood 

.9000 

508.6  ' 

Steam  at  212°  F 

.4880 

275.8 

The  weight  of  a cubic  foot  of  any  solid  or  liquid  is  found 
by  multiplying  its  specific  gravity  by  62.425  lb.  avoirdupois. 
The  weight  of  a cubic  foot  of  any  gas  at  atmospheric  pres- 
sure and  at  32°  F.  is  found  by  multiplying  its  specific  gravity 
by  .08073  lb.  avoirdupois. 


WROUGHT-IRON  CHAIN  CABLES. 

The  strength  of  a chain  link  is  less  than  twice  that  of  a 
straight  bar  of  a sectional  area  equal  to  that  of  one  side  of  the 
link.  A weld  exists  at  one  end  and  a bend  at  the  other,  each 
requiring  at  least  one  heat,  which  produces  a decrease  in  the 
strength.  The  report  of  the  committee  of  the  U.  S.  Testing 
Board,  on  tests  of  wrought-iron  and  chain  cables,  contains 
the  following  conclusions: 

“That  beyond  doubt,  when  made  of  American  bar  iron, 
with  cast-iron  studs,  the  studded  link  is  inferior  in  strength 
to  the  unstudded  one. 


CHAIN  CABLES. 


15 


“That,  when  proper  care  is  exercised  in  the  selection  of 
material,  a variation  of  50  to  170  of  the  strongest  may  be 
expected  in  the  resistance  of  cables.  Without  this  care  the 
variation  may  rise  to  250. 

“ That  with  proper  material  and  construction  the  ultimate 
resistance  of  the  chain  may  be  expected;to  vary  from  1550  to 
1700  of  that  of  the  bar  used  in  making  the  links,  and  show  an 
average  of  about  1630. 

“That  the  proof  test  of  a chain  cable  should  be  about  500 
of  the  ultimate  resistance  of  the  weakest  link.” 

From  a great  number  of  tests  of  bars  and  unfinished 
cables,  the  committee  considered  that  the  average  ultimate 
resistance  and  proof  tests  of  chain  cables  made  of  the  bars, 
whose  diameters  are  given,  should  be  such  as  are  shown  in 
the  accompanying  table. 

Ultimate  Resistance  and  Proof  Tests  of  Chain  Cables. 


Diam. 

of 

Bar. 

Inches. 

Average 
Resist.  = 
1630  of  Bar. 

Pounds. 

Proof 

Test. 

Pounds. 

Diam. 

of 

Bar. 

Inches. 

Average 
Resist.  = 
1630  of  Bar. 

Pounds. 

Proof 

Test. 

Pounds. 

1 

71,172 

33,840 

1* 
i k 
ih 

162,283 

77,159 

\p. 

79,544 

37,820 

174,475 

82,956 

88,445 

42,053 

187,075 

88,947 

IP 

97,731 

46,468 

200,074 

95,128 

107,440 

51,084 

! 

lit 

213,475 

101,499 

V* 

117,577 

55,903 

227,271 

108,058 

1% 

128,129 

60,920 

241,463 

114,806 

IP 

139,103 

150,485 

66,138 

71,550 

2 

256,040 

121,737 

TYPE  METALS. 

Name.  Proportions. 

Smallest  type  3 X,  1 A 

Small  type 4 1 A 

Medium  type  5 X,  1 A 

Large  type 6 X,  1 A 

Largest  type 7 X,  1 A 

In  the  above  table,  X represents  the  lead,  and  A the  anti- 
mony in  the  alloy. 


16 


USEFUL  TABLES. 


TABLE  OF  ELEMENTS. 


Aluminum 

Antimony  (stibium) 

Arsenic  

Barium 

Beryllium  

Bismuth 

Boron 

Bromine 

Cadmium  

Caesium  

Calcium  

Carbon  

Cerium 

Chlorine 

Chromium 

Cobalt  

Columbium  

Copper  (cuprum)  

Didymium 

Erbium  

Fluorine 

Gallium  

Germanium  

Gold  (aurum)  

Hydrogen  

Indium  

Iodine 

Iridium  

Iron  (ferrum)  

Lanthanum  

Lead  (plumbum)  

Lithium  

Magnesium  

Mercury  (hydrargyrum) 

Manganese....'. 

Molybdenum 

Nickel  

Niobium 

Nitrogen  

Osmium  

Oxygen  


rmbol. 

Atomic 

Weight/ 

Al 

27.04 

Sb 

119.96 

As 

74.9 

Ba 

136.9 

Be 

9.08 

Bi 

207.5 

B 

10.9 

Br 

79.76 

Cd 

111.7 

Cs 

133.0 

Ca 

39.91 

C 

11.97 

Ce 

141.2 

Cl 

35.37 

Cr 

52.45 

Co 

58.6 

Cb 

93.7 

Cu 

63.18 

D 

147.0 

E 

169.0 

F 

19.06 

G 

69.8 

Ge 

72.32 

Au 

196.2 

H 

1.0 

In 

113.4 

I 

126.54 

Ir 

196.7 

Fe 

55.88 

Lqi 

139.0 

Pb 

206.39 

Li 

7.01 

Mg 

23.94 

Mg 

199.8 

Mn 

54.8 

Mo 

95.6 

Ni 

58.6 

94.0 

14.01 

Os 

198.6 

0 

15.96 

* Principally  from  the  16th  edition  Des  Ingenieurs  Taschen- 
buch.  The  names  of  the  non-metals  are  printed  in  heavy  type. 


TABLE  OF  SPECIFIC  HEATS. 


17 


Table — ( Continued). 


Symbol. 


Atomic 

Weight. 


Palladium 

Phosphorus  

Platinum 

Potassium  (kalium) 

Rhodium 

Rubidium  

Ruthenium  

Scandium  

Selenium  

Silicon  

Silver  (argentum)  ... 
Sodium  (natrium) ... 

Strontium  

Sulphur 

Tantalum  

Tellurium  

Thallium 

Thorium 

Tin  (stannum) 

Titanium 

Tungsten  (wolfram) 

Uranium 

Vanadium  

Ytterbium 

Yttrium  

Zinc 

Zirconium 


Pd 

106.2 

P 

30.96 

Pt 

194.43 

K 

39.04 

Rh 

104.1 

Rb 

85.2 

Ru 

103.5 

Sc 

44.04 

Se 

78.00 

Si 

28.00 

Ag 

107.66 

Na 

23.0 

Sr 

87.3 

S 

31.98 

Ta 

182.0 

Te 

128.0 

Tl 

203.6 

Th 

231.5 

Sn 

117.35 

Ti 

48.0 

W 

183.6 

U 

240.0 

V 

51.2 

Yb 

93.0 

Y 

172.6 

Zn 

64.88 

Zr 

90.0 

TABLE  OF  SPECIFIC  HEATS. 


Solids. 


Copper 0951 

Cast  iron 1298 

Gold 0324 

Wrought  ir,on  1138 

Lead 0314 

Platinum  0324 

Steel  (soft) 1165 

Steel  (hard)  1175 

Zinc 0956 

Silver  0570 

Tin  0562 

Ice  5040 

Brass  .0939 

Glass 1937 

Sulphur  2026 

Charcoal 2410 

18 


USEFUL  TABLES. 


Liquids. 


Water  1.0000 

Alcohol  7000 

Mercury  0333 

Benzine  4500 

Glycerine .5550 


Lead  (melted) 0402 

Sulphur  (melted)  2340 

Tin  (melted) 0637 

Sulphuric  acid  3350 

Oil  of  turpentine  4260 


Gases. 


Air 

23751 

Superheated  steam  .... 

...  .4805 

Oxygen  

21751 

Carbonic  oxide  (CO).... 

..  .2479 

Nitrogen 

24380 

Carbonic  acid  ( C02 ) ■■■■ 

...  .2170 

Hydrogen 

3.40900 

Olefiant  gas  

...  .4040 

TEMPERATURES  AND  LATENT  HEATS  OF  FUSION 
AND  OF  VAPORIZATION. 


Substance. 

Temperature 

of 

Fusion. 

Temperature 

of 

Vaporization. 

Latent  Heat 
of 

Fusion. 

Latent  Heat 
of 

Vaporization. 

Water 

32° 

212° 

142.65 

966.6 

Mercury  

-37.8° 

662° 

5.09 

157 

Sulphur. 

228.3° 

824° 

13.26 

Tin 

446° 

25.65 

Lead  

626° 

9.67 

Zinc  

680° 

1,900° 

50.63 

493 

Alcohol 

Unknown 

173° 

372 

Oil  of  turpentine ... 

14° 

313° 

124 

Linseed  oil 

600° 

Aluminum 

1,400° 

Copper  

2,100° 

Cast  iron 

2,192° 

3,300° 

Wrought  iron  

2,912° 

5,000° 

Steel  

2,520° 

Platinum 

3,632° 

Iridium 

4,892° 

Example.— How  many  units  of  heat  are  required  to  melt 
10  lb.  of  zinc  from  a temperature  of  60°  F.? 


COEFFICIENTS  OF  EXPANSION. 


19 


Solution.— The  specific  heat  of  zinc  is  found  from  the 
table  to  be  .0956.  Hence,  the  number  of  heat  units  necessary 
to  raise  it  to  the  melting  point  is  10  X (680  — 60)  X .0956 
= 592.72.  Latent  heat  of  fusion  = 50.63  heat  units.  Hence, 
the  total  number  of  heat  units  required  is  592.72  + 10  X 50.63 
= 1,099.02.  


HEAT. 

Coefficient  of  Expansion  for  a Number  of  Substances. 


Name  of  Substance. 

Linear 

Expansion. 

Surface 

Expansion. 

Cubic 

Expansion. 

Cast  iron  

.00000617 

.00001234 

.00001850 

Copper  

.00000955 

.00001910 

.00002864 

Brass  

.00001037 

.00002074 

.00003112 

Silver 

.00000690 

.00001390 

.00002070 

Bar  iron 

.00000686 

.00001372 

.00002058 

Steel  (untempered) 

.00000599 

.00001198 

.00001798 

Steel  (tempered)  

.00000702 

.00001404 

.00002106 

Zinc 

.00001634 

.00003268 

.00004903 

Tin  

.00001410 

.00002820 

.00003229 

Mercury 

.00003334 

.00006668 

.00010010 

Alcohol 

.00019259 

.00038518 

.00057778 

Gases  

.00203252 

Example.— A wrought-iron  bar  22  ft.  long  is  heated  from 
70°  to  300°.  How  much  will  it  lengthen? 

Solution—  22  X (300  - 70)  X .00000686  = .0347116  ft.  = 
.41654  in.  


ALLOYS. 

Note. — A = Antimony,  B = Bismuth,  C = Copper,  G = 
Gold,  I = Iron,  L = Lead,  N = Nickel,  S = Silver,  T = Tin, 
Z = Zinc. 

Name.  Proportions. 

Brass,  common  yellow 2 C,  1 Z 

Brass,  to  be  rolled 32  C,  10  Z,  1.5  T 

Brass  castings,  common  ...  20  C,  1.25  Z,  2.5  T 

Brass  castings,  hard 25  C,  2 Z,  4.5  T 

Brass  propellers  8 C,  .5  Z,  1 T 

Gun  metal  8 C,  1 T 


20 


USEFUL  TABLES. 


Alloys — ( Continued). 

Name.  Proportions. 

Copper  flanges 9 C,  1 Z,  .26  T 

Muntz’s  metal 6 C,  4Z 

Statuary  91.4  C,  5.53  Z,  1.7  T,  1.37  L 

German  silver  2 C , 7.9  N,  6.3  Z,  6.5 1 

Britannia  metal 50  A,  25  T,  25  B 

Chinese  silver  65.1  C,  19.3  Z,  13  N,  2.58  S , 12 1 

Chinese  white  copper  20.2  C,  12.7  Z,  1.3  T,  15.82V 

Medals 100  C,  8 Z 

Pinchbeck  . 5 C,  1 Z 

Babbitt’s  metal 25  T,  2 A,  .5  C 

Bell  metal,  large  3 C,  1 T 

Bell  metal,  small 4 Cf  1 T 

Chinese  gongs  40.5  C,  9.2  T 

Telescope  mirrors  33.3  C , 16.7  T 

White  metal,  ordinary 3.7  C,  3.7  Z,  14.2  T,  28.4  A 

White  metal,  hard 35  C,  13  Z,  2.2  T 

Sheeting  metal 56  C,  45  Z,  12  arsenic 

Metal,  expands  in  cooling  75  L,  16.7  A,  8.3  B 


ALLOYS  FOR  SOLDERS. 

Name.  Proportions. 


Melting 

Point. 


Newton’s  fusible 8 B,  5 L,  3 T,  212° 

Rose’s  fusible 2 B,  1 L,  1 T,  201° 

A more  fusible 5 B,  3 L,  2 T,  199° 

Still  more  fusible 12  T,  25 L,  50  B,  13  cadmium,  155° 

For  tin  solder,  coarse, 1 T,  3 X,  500° 

For  tin  solder,  ordinary 2 T,  1 L,  360° 

For  brass,  soft  spelter 1 C,  1 Z,  550° 

B ard,  for  iron  2 C,  1 Z,  700° 

1 or  steel  19  S,  3 C,  1 Z 

For  fine  brasswork  1 S,  8 C,  8Z 

Pewterer’s  soft  solder 2 B,  4X,  3 T 

Pewterer’s  soft  solder 1 B,  1 L,  2 T 

Gold  solder 24  G,  2S,1C 

Silver  solder,  hard  4 S,  1 C 

Silver  solder,  soft 2 S,  1 brass  wire 

For  lead  16  T,  33  X 


ROLLED  IRON. 


21 


WEIGHT  OF  ROUND  AND  SQUARE  ROLLED  IRON. 

From  ^ in.  to  9%  in-  in  Diameter,  and  1 ft.  in  Length. 


Side  or 
Diam. 
Inches. 

Weight.  Lb.  per  ft. 

Side  or 
Diam. 
Inches. 

Weight.  Lb.  per  ft. 

Round. 

Square. 

Round. 

Square. 

i h 

.010 

.013 

3% 

39.864 

50.756 

% 

.041 

.053 

4 

42.464 

54.084 

.093 

.118 

4% 

45.174 

57.517 

k 

.165 

.211 

4 y 

47.952 

61.055 

s| 

.373 

.475 

4% 

50.815 

64.700 

.663 

.845 

4*4 

53.760 

68.448 

% 

1.043 

1.320 

4^1 

56.788 

72.305 

7a 

1.493 

1.901 

59.900 

76.264 

/8 

2.032 

2.588 

m 

63.094 

80.333 

1 

2.654 

3.380 

5 

66.350 

84.480 

V/s 

3.359 

4.278 

5% 

69.731 

88.784 

V4 

4.147 

5.280 

534 

73.172 

93.168 

1% 

5.019 

6.390 

5% 

76.700 

97.657 

5.972 

7.604 

534 

80.304 

102.240 

i% 

7.010 

8.926 

5% 

84.001 

106.953 

■ M 

8.128 

10.352 

534 

87.776 

111.756 

9.333 

11.883 

91.634 

116.671 

2 

10.616 

13.520 

6 

95.552 

121.664 

2% 

11.988 

15.263 

103.704 

132.040 

2/i 

13.440 

17.112 

112.160 

142.816 

2^1 

14.975 

19.066 

6?| 

120.960 

154.012 

16.588 

21.120 

7 

130.048 

165.632 

2tI 

18.293 

23.292 

7^ 

139.544 

177.672 

2M 

20.076 

25.560 

7>| 

149.328 

190.136 

2% 

21.944 

27.939 

7^ 

159.456 

203.024 

3 

23.888 

30.416 

8 

169.856 

216.336 

3% 

25.926 

33.010 

834 

180.696 

230.068 

3K 

28.040 

35.704 

8/1 

191.808 

244.220 

n 

30.240 

38.503 

3/4 

203.260 

258.800 

32.512 

41.408 

9 

215.0-iO 

273.792 

3P 

34.886 

44.418 

227.152 

289.220 

3/4 

37.332 

47.534 

9>1 

239.600 

305.056 

WEIGHT  OF  SHEET  LEAD. 


Thickness. 

Inches. 

W£ht. 

Thickness. 

Inches. 

W’ght. 

Lb. 

Thickness. 

Inches. 

W’ght. 

.017 

1 

.085 

5 

.152 

9 

.034 

2 

.101 

6 

.169 

10 

.051 

3 

.118 

7 

.186 

11 

.068 

4 

.135 

8 

.203 

• 12 

PROPORTIONS  OF  THE  UNITED  STATES  STANDARD 
SCREW  THREADS,  NUTS,  AND  BOLT  HEADS. 


Notation  of  letters.  All  dimensions  in  inches. 


D — outside  diameter  of  screw;. 
d = diameter  of  root  of  thread,  or  of 
hole  in  the  nut; 
p = pitch  of  screw; 
t = number  of  threads  per  inch; 

/ = flat  top  and  bottom; 

© = outside  diameter  of  hexagon  nut 
or  bolt  head; 


< = inside  [diameter  of  hexagon,  or 
side  of  square  nut  or  bolt  head; 
s = diagonal  of  square  nut  or  bolt 
head; 

h = height  of  rough  or  unfinished  bolt 
head. 

The  height  of  finished  nut  or  bolt 
head  is  made  equal  to  the  diameter  D 
of  the  screw. 


V = 


vu 


d = D- 


D + 10  — 2.909  1 

16.64  ' ~ p 

1.299  . 3 Z>  1 

“•  l = ^r  + 8-  0= 


t = 1.414  i. 
1.155  i.  /=|. 


CAST-IRON  PIPE. 


23 


WEIGHT  OF  CAST-IRON  PIPE  PER  FOOT  IN  POUNDS. 

These  weights  are  for  plain  pipe.  For  hautboy  pipe  add 
8 in.  in  length  for  each  joint.  For  copper  add  for  lead,  g ; 
for  welded  iron,  add  TV,  or  multiply  by  1.0667. 


Thickness  of  Pipe  in  Inches. 


liore. 

Inches. 

34 

% 

'X 

% 

% 

Vs 

1 

V4 

1% 

1 

3.07 

5.07 

7.38 

1M 

3.69 

6.00 

8.61 

4.30 

6.92 

9.84 

1% 

4.92 

7.84 

11.10 

2 

5.53 

8.76 

12.30 

16.2 

2 M 

6.15 

9.69 

13.50 

17.7 

20 

6.76 

10.60 

14.80 

19.2 

24.0 

2/4 

7.37 

11.50 

16.00 

20.8 

25.9 

3 

7.98 

12.50 

17.20 

22.3 

27.7 

33.4 

3 % 

9.21 

14.30 

19.70 

25.4 

31.4 

37.7 

4 

10.30 

16.10 

22.20 

28.5 

35.1 

42.0 

4 % 

11.70 

18.00 

24.60 

31.5 

38.8 

46.3 

5 

12.90 

19.80 

27.10 

34.6 

42.5 

50.6 

5% 

14.20 

21.70 

29.50 

37.7 

46.1 

54.9 

6 

15.40 

23.50 

32.00 

40.8 

49.8 

59.2 

68.9 

6% 

16.60 

25.40 

34.50 

43.8 

53.5 

63.5 

73.8 

84.4 

7 

17.80 

27.20 

36.90 

46.9 

57.2 

67.8 

78.7 

89.4 

7^ 

19.10 

29.10 

39.40 

50.0 

60.9 

72.1 

83.7 

95.5 

108 

8 

20.30 

30.90 

41.80 

53.1 

64.6 

76.4 

88.6 

101.0 

114 

127 

8% 

21.50 

32.80 

44.30 

56.1 

68.3 

80.7 

93.5 

107.0 

120 

134 

9 

22.80 

34.60 

46.80 

59.2 

72.0 

85.1 

98.4 

112.0 

126 

140 

9 >4 

24.00 

36.40 

49.20 

62.3 

75.7 

89.3 

103.0 

118.0 

132 

147 

10 

25.10 

38.30 

51.70 

65.3 

79.4 

93.6 

108.0 

123.0 

138 

164 

11 

27.60 

42.00 

56.60 

71.5 

86.7 

102.0 

118.0 

134.0 

151 

168 

12 

30.00 

45.70 

61.50 

77.7 

94.1 

111.0 

128.0 

145.0 

163 

181 

13 

32.50 

49.40 

66.40 

83.8 

102.0 

120.0 

138.0 

156.0 

175 

195 

14 

35.00 

53.10 

71.40 

89.4 

109.0 

128.0 

148.0 

168.0 

188 

208 

15 

37.40 

56.70 

76.30 

96.1 

116.0 

137.0 

158.0 

179.0 

200 

222 

16 

39.10 

60.40 

81.20 

102.0 

124.0 

145.0 

167.0 

190.0 

212 

235 

17 

42.30 

64.10 

86.10 

108.0 

131.0 

154.0 

177.0 

201.0 

225 

249 

18 

44.80 

67.80 

91.00 

115.0 

139.0 

163.0 

187.0 

212.0 

237 

262 

19 

47.30 

71.50 

96.00 

121.0 

146.0 

171.0 

197.0 

223.0 

249 

276 

20 

49.70 

75.20 

101.00 

127.0 

153.0 

180.0 

207.0 

234.0 

261 

289 

22 

54.60 

82.60 

111.00 

139.0 

168.0 

196.0 

227.0 

256.0 

286 

316 

24 

59.60 

89.90 

121.00 

152.0 

183.0 

214.0 

246.0 

278.0 

311 

343 

26 

64.50 

97.30 

131.00 

164.0 

198.0 

231.0 

266.0 

300.0 

335 

370 

28 

69.40 

105.00 

140.00 

176.0 

212.0 

249.0 

286.0 

323.0 

360 

397 

30 

74.20 

112.00 

150.00 

188.0 

227.0 

266.0 

305.0 

345.0 

384 

424 

24 


USEFUL  TABLES. 


TABLE  OF  STANDARD  DIMENSIONS  OF  WROUGHT- 
IRON  WELDED  PIPES. 


Nominal  Diameter. 

External  Diameter. 

Thickness. 

Internal  Diameter. 

I 

Internal  Circum- 
ference. 

External  Circum- 
ference. 

Length  of  Pipe  per 
Sq.  Ft.  of  Inter- 
nal Surface. 

Length  of  Pipe  per 
Sq.  Ft.  of  Exter- 
nal Surface. 

Internal  Area. 

Weight  per  Foot. 

No.  of  Threads  per 
Inch  of  Screw. 

In. 

In. 

In. 

In. 

In. 

In. 

Ft. 

Ft. 

In. 

Lb. 

Vs 

.40 

.068 

.27 

.85 

1.27 

14.15 

9.440 

.057 

.24 

27 

17 

.54 

.088 

.36 

1.14 

1.70 

10.50 

7.075. 

.104 

.42 

18 

78 

.67 

.091 

.49 

1.55 

2.12 

7.67 

5.657 

.192 

.56 

18 

.84 

.109 

.62 

1.96 

2.65 

6.13 

4.502 

.305 

.84 

14 

/A 

1.05 

.113 

.82 

2.59 

3.30 

4.64 

3.637 

.533 

1.13 

14 

1 

1.31 

.134 

1.05 

3.29 

4.13 

3.66 

2.903 

.863 

1.67 

IK 

1.66 

.140 

1.38 

4.33 

5.21 

2.77 

2.301 

1.496 

2.26 

11% 

i % 

1.90 

.145 

1.61 

5.06 

5.97 

2.37 

2.010 

2.038 

2.69 

11% 

2 

2.37 

.154 

2.07 

6.49 

7.46 

1.85 

1.611 

3.355 

3.67 

11% 

2% 

2.87 

.204 

2.47 

7.75 

9.03 

1.55 

1.328 

4.783 

5.77 

8 

3 

3.50 

.217 

3.07 

9.64 

11.00 

1.24 

1.091 

7.388 

7.55 

8 

4.00 

.226 

3.55 

11.15 

12.57 

1.08 

0.955 

9.887 

9.05 

8 

4 

4.50 

.237 

4.03 

12.65 

14.14 

.95 

0.849 

12.730 

10.73 

8 

4^ 

5.00 

.247 

4.51 

14.15 

15.71 

.85 

0.765 

15.939 

12.49 

8 

5 

5.56 

.259 

5.04 

15.85 

17.47 

.78 

0.629 

19.990 

14.56 

8 

6 

6.62 

.280 

6.06 

19.05 

20.81 

.63 

0.577 

28.889 

18.77 

8 

7 

7.62 

.301 

7.02 

22.06 

23.95 

.54 

0.505 

38.737 

23.41 

8 

8 

8.62 

.322 

7.98 

25.08 

27.10 

.48 

0.444 

50.039 

28.35 

8 

9 

9.69 

.344 

9.00 

28.28 

30.43 

.42 

0.394 

63.633 

34.08 

8 

10 

10.75 

.366 

10.02 

i 

31.47 

33.77 

.38 

0.355 

78.838 

40.64 

8 

FLUXES  FOR  SOLDERING  OR  WELDING. 


Iron  Borax 

Tinned  iron  Resin 


Copper  and  brass 

Sal  ammoniac 


Zinc Chloride  of  zinc 

Lead  Tallow  or  resin 


Lead  and  tin  pipes 

Resin  and  sweet  oil 


Steel.— Pulverize  together  1 part  of  sal  ammoniac  and  10 
parts  of  borax  and  fuse  until  clear.  When  solidified,  pul- 
verize to  powder. 


STEAM  TABLES. 


25 


STEAM  TABLES. 

Whenever  the  pressure  of  saturated  steam  is  changed, 
there  are  other  properties  that  change  with  it.  These  prop- 
erties are  the  following: 

1.  The  temperature  of  the  steam,  or,  what  is  the  same 
thing,  the  boiling  point. 

2.  The  number  of  B.  T.  U.  required  to  raise  a pound  of 
water  from  32°  (freezing)  to  the  boiling  point  corresponding 
to  the  given  pressure.  This  is  called  the  heat  of  the  liquid. 

3.  The  number  of  B.  T.  U.  required  to  change  the  water 
at  the  boiling  temperature  into  steam  at  the  same  tempera- 
ture. This  is  called  the  latent  heat  of  vaporization , or,  simply, 
the  latent  heat. 

4.  The  number  of  heat  units  required  to  change  a pound 
of  water  at  32°  to  steam  of  the  required  temperature  and 
pressure.  This  is  called  the  total  heat  of  vaporization , or, 
simply,  the  total  heat. 

It  is  plain  that  the  total  heat  is  the  sum  of  the  heat  of  the 
liquid  and  the  latent  heat.  That  is,  total  heat  = heat  of 
liquid  + latent  heat. 

5.  The  specific  volume  of  the  steam  at  the  given  pressure; 
that  is,  the  number  of  cubic  feet  occupied  by  a pound  of 
steam  of  the  given  pressure. 

6.  The  density  of  the  steam;  that  is,  the  weight  of  1 cubic 
foot  of  the  steam  at  the  given  pressure. 

All  the  above  properties  are  different  for  different  pres- 
sures. For  example,  if  steam  boils  under  atmospheric  pres- 
sure, the  temperature  is  212°;  the  heat  of  the  liquid  is  180.531 
B.  T.  U.;  the  latent  heat,  966.069  B.  T.  U.;  the  total  heat, 
1,146.6  B.  T.  U.  A pound  of  steam  at  this  pressure  occupies 
26.37  cu.  ft.,  and  a cubic  foot  of  the  steam  weighs  about 
.037928  lb.  When  the  pressure  is  70  lb.  per  sq.  in.  above' 
vacuum,  the  temperature  is  302.774°;  the  heat  of  the  liquid  is 
272.657  B.  T.-U.;  the  latent  heat  is  901.629  B.  T.  U.;  the  total 
heat  is  1,174.286  B.  T.  U.  A pound  of  the  steam  occupies  6.076 
cu.  ft.,  and  a cubic  foot  of  the  steam  weighs  .164584  lb. 

These  properties  have  been  determined  by  direct  experi- 
ment for  all  ordinary  steam  pressures.  They  are  given  in 
the  table  of  the  properties  of  saturated  steam,  pages  29-31. 


26 


USEFUL  TABLES. 


Explanation  of  the  Table. 

Column  1 gives  the  pressures  from  1 to  300  lb.  These  pres- 
sures are  above  vacuum.  The  steam  gauges  fitted  on  steam 
boilers  register  the  pressure  above  the  atmosphere.  That  is, 
if  the  steam  is  at  atmospheric  pressure,  14.7  lb.  per  sq.  in.,  the 
gauge  registers  0.  Consequently,  the  atmospheric  pressure 
must  be  added  to  the  reading  of  the  gauge  to  obtain  the  pres- 
sure above  vacuum.  In  using  the  table,  care  must  be  taken 
not  to  use  the  gauge  pressures  without  first  adding  14.7  lb. 
per  sq.  in. 

Pressures  registered  above  vacuum  are  called  absolute 
pressures.  The  pressures  given  in  column  1 are  absolute. 
Absolute  pressure  per  square  inch  = gauge  pressure  per 
square  inch  -f  14.7. 

Column  2 gives  the  temperature  of  the  steam  when  at  the 
pressure  shown  in  column  1. 

Column  3 gives  the  heat  of  the  liquid.  It  will  be  noticed 
that  the  values  in  column  3 may  be  obtained  approximately 
by  subtracting  32°  from  the  temperature  in  column  2.  If  the 
specific  heat  of  water  were  exactly  1.00,  it  would,  of  course, 
take  exactly  212  — 32  = 180  B.  T.  U.  to  raise  a pound  of  water 
from  32°  to  212°.  But  experiment  shows  that  the  specific 
heat  of  water  is  slightly  greater  than  1.00  when  the  temper- 
ature of  the  water  is  above  62°,  and  it  therefore  takes  180.531 
B.  T.  U.  to  raise  a pound  of  water  from  32°  to  212°. 

Column  4 gives  the  latent  heat  of  vaporization , which  is  seen 
to  decrease  slightly  as  the  pressure  increases. 

Column  5 gives  the  total  heat  of  vaporization.  The  values 
in  column  5 may  be  obtained  by  adding  together  the  corre- 
sponding values  in  columns  3 and  4. 

Column  6 gives  the  weight  of  a cubic  foot  of  steam  in 
pounds.  As  would  be  expected,  the  steam  becomes  denser 
as  the  pressure  rises,  and  weighs  more  per  cubic  foot. 

Column  7 gives  the  number  of  cubic  feet  occupied  by  1 
pound  of  steam  at  the  given  pressure.  It  will  be  noticed  that 
the  corresponding  values  of  columns  6 and  7 multiplied 
together  always  produce  1.  Thus,  for  31.3  pounds  pressure, 
gauge,  .11088  X 9.018  = 1.000,  nearly. 

Column  8 gives  the  ratio  of  the  volume  of  a pound  of 


STEAM  TABLES. 


27 


steam  at  the  given  pressure,  and  the  volume  of  a pound  of 
water  at  39.2°.  The  values  in  column  8 may  be  obtained  by 
dividing  62.425,  the  weight  of  a cubic  foot  of  water  at  39.2°, 
by  the  numbers  in  column  6. 

Examples  on  the  Use  of  the  Steam  Table. 

Example  1.— Calculate  the  heat  required  to  change  5 lb. 
of  water  at  32°  into  steam  at  92  lb.  pressure  above  vacuum. 

Solution.— From  column  5,  the  total  heat  of  1 lb.  at  92  lb. 
pressure  is  1,180.045  B.  T.  U. 

1,180.045  X 5 = 5,900.225  B.  T.  U. 

Example  2.— How  many  heat  units  are  required  to  raise 

lb.  of  water  from  32°  to  250°  F.? 

Solution.— Looking  in  column  3,  the  heat  of  the  liquid 
of  1 lb.  at  250.293°  is  219.261  B.  T.  U.  219.261  — .293  = 218.968 
B.  T.  U.  = heat  of  liquid  for  250°.  Then,  for  8£  lb.  it  is 
218.968  X 84-  = 1,861.228  B.  T.  U. 

Example  3.— How  many  foot-pounds  of  work  will  it 
require  to  change  60  lb.  of  boiling  water  at  80  lb.  pressure, 
absolute,  into  steam  of  the  same  pressure? 

Solution.— Looking  under  column  4,  the  latent  heat  of 
vaporization  is  895.108;  that  is,  it  takes  895.108  B.  T.  U.  to 
change  1 lb.  of  water  at  80  lb.  pressure  into  steam  of  the  same 
pressure.  Therefore,  it  takes  895.108  X 60  = 53,706.48  B.  T.  U. 
to  perform  the  same  operation  on  60  lb.  of  water. 

53,706.48  X 778  = 41,783,641.44  ft.-lb. 

Example  4.— Find  the  volume  occupied  by  14  lb.  of  steam 
at  30  lb.,  gauge  pressure. 

Solution. — 30  lb.,  gauge  pressure  = 30  + 14.7  = 44.7,  abso- 
lute pressure.  The  nearest  pressure  in  the  table  is  44  lb.,  and 
the  volume  of  a pound  of  steam  at  that  pressure  is  9.403  cu.  ft. 
The  volume  of  a pound  at  46  lb.  pressure  is  9.018  cu.  ft. 
9.403  — 9.018  = .385  cu.  ft.,  the  difference  in  volume  for  a 
385 

difference  in  pressure  of  2 lb.  = .1925  cu.  ft.,  the  differ- 
ence in  volume  for  a difference  in  pressure  of  1 lb.  .1925  X .7 
= .135  cu.  ft.,  the  difference  in  volume  for  a difference  in 
pressure  of  .7  lb.  Therefore,  9.403  — .135  = 9.268  cu.  ft.  is  the 
volume  of  1 lb.  of  steam  at  44.7  lb.  pressure.  The  .135  cu.  ft. 


28 


USEFUL  TABLES. 


is  subtracted  from  9.403  cu.  ft.,  since  the  volume  is  less  for 
a pressure  of  44.7  lb.  than  for  a pressure  of  44  lb. 

9.268X14  = 129.752  cu.  ft. 

Example  5.— Find  the  weight  of  40  cu.  ft.  of  steam  at  a 
temperature  of  254°  F. 

Solution.— The  weight  of  1 cu.  ft.  of  steam  at  254.002°, 
from  the  table,  is  .078839  lb.  Neglecting  the  .002°,  the  weight 
of  40  cu.  ft.  is,  therefore, 

.078839  X 40  = 3.15356  lb. 

Example  6.— How  many  pounds  of  steam  at  64  lb.  pressure, 
absolute,  are  required  to  raise  the  temperature  of  300  lb.  of 
water  from  40°  to  130°  F.,  the  water  and  steam  being  mixed  ? 

Solution.— The  number  of  heat  units  required  to  raise 
1 lb.  from  40°  to  130°  is  130  — 40  = 90  B.  T.  U.  (Actually  a 
little  more  than  90  would  be  required,  but  the  above  is  near 
enough  for  all  practical  purposes.)  Then,  to  raise  300  lb. 
from  40°  to  130°  requires  90  X 300  = 27,000  B.  T.  U.  This 
quantity  of  heat  must  necessarily  come  from  the  steam. 
Now,  1 lb.  of  steam  at  64  lb.  pressure  gives  up,  in  condensing, 
its  latent  heat  of  vaporization,  or  905.9  B.  T.  U.  But,  in  addi- 
tion to  its  latent  heat,  each  pound  of  steam  on  condensing 
must  give  up  an  additional  amount  of  heat  in  falling  to  130°. 
Since  the  original  temperature  of  the  steam  was  296.805°  F. 
(see  table),  each  pound  gives  up  by  its  fall  of  temperature 
296.805  — 130  = 166.805  B.  T.  U.  Therefore,  each  pound  of  the 
steam  gives  up  a total  of 

905.9  + 166.805  = 1,072.705  B.  T.  U. 

It  will,  therefore,  take  = 25.17  lb.  of  steam  to 

1,072.705 

accomplish  the  desired  result. 

With  the  steam  tables  a reliable  thermometer  may  be  used 
for  ascertaining  the  pressure  of  saturated  steam  or  for  testing 
the  accuracy  of  a steam  gauge.  The  temperature  of  the  steam 
being  measured  by  the  thermometer,  the  corresponding  abso- 
lute pressure  is  found  from  the  steam  tables;  the  gauge  pres- 
sure is  then  found  by  subtracting  14.7  from  the  absolute 
pressure.  Thus,  the  temperature  of  the  steam  in  a condenser 
being  142°,  we  find  from  the  steam  tables  that  the  correspond- 
ing absolute  pressure  is  3 lb.  per  sq.  in.,  nearly. 


STEAM  TABLES. 


29 


The  Properties  of  Saturated  Steam. 


Pressure  Above  Vacuum  in 
Pounds  per  Square  Inch. 

Temperature,  Fahrenheit 
Degrees. 

Quantity  of  Heat  in 
British  Thermal  Units. 

Weight  of  a Cubic  Foot  of  Steam 
in  Pounds. 

Volume  of  a Pound  of  Steam  in 
Cubic  Feet. 

Ratio  of  Vol.  of  Steam  to  Vol.  of 
Equal  Weight  of  Dist.  Water  at 
Temp,  of  Maximum  Density. 

Required  to  Raise  Tem- 
perature of  the  Water 
From  32°  to  t°. 

Total  Latent  Heat  at 
Pressure  p. 

Total  Heat  Above  32°. 

1 

2 

3 

4 

5 

6 

7 

8 

P 

t 

Q 

L 

H 

W 

V 

R 

1 

102.018 

70.040 

1,043.015 

1,113.055 

.003027 

330.4 

20,623 

2 

! 126.302 

94.368 

1,026.094 

1,120.462 

.005818 

171.9 

10,730 

3 

: 141.654 

109.764 

1,015.380 

1,125.144 

.008522 

117.3 

7,325 

4 

153.122 

121.271 

1,007.370 

1,128.641 

.011172 

89.51 

5,588 

5 

162.370 

130.563 

1,000.899 

1,131.462 

.013781 

72.56 

4,530 

6 

170.173 

138.401 

995.441 

1,133.842 

.016357 

61.14 

3,816 

7 

176.945 

145.213 

990.695 

1,135.908 

.018908 

52.89 

3,302 

8 

182.952 

151.255 

986.485 

1,137.740 

.021436 

46.65 

2,912 

9 

188.357 

156.699 

982.690 

1,139.389 

.023944 

41.77 

2,607 

10 

193.284 

161.660 

979.232 

1,140.892 

.026437 

37.83 

2,361 

11 

197.814 

166.225 

976.050 

1,142.275 

.028911 

34.59 

2,159 

12 

1 202.012 

170.457 

973.098 

1,143.555 

.031376 

31.87 

1,990 

13 

I 205.929 

174.402 

970.346 

1,144.748 

.033828 

29.56 

1,845 

14 

209.604 

178.112 

967.757 

1,145.869 

.036265 

27.58 

1,721 

14.69 

: 212.000 

180.531 

966.069 

1,146.600 

.037928 

26.37 

1,646 

15 

213.067 

181.608 

965.318 

1,146.926 

.038688 

25.85 

1,614 

16 

j 216.347 

184.919 

963.007 

1.147.926 

.041109 

24.33 

1,519 

17 

! 219.452 

188.056 

960.818 

1,148.874 

.043519 

22.98 

1,434 

18 

222.424 

191.058 

958.721 

1,149.779 

.045920 

21.78 

1,359 

19 

j 225.255 

193.918 

956.725 

1,150.613 

.048312 

20.70 

1,292 

30 


USEFUL  TABLES. 


Table— ( Continued). 


1 

2 

3 

4 

5 

6 

7 

8 

V 

t 

Q 

L 

H 

W 

V 

It 

20 

227.964 

196.655 

954.814 

1,151.469 

.050696 

19.730 

1,231.0 

22 

233.069 

201.817 

951.209 

1,153.026 

.055446 

18.040 

1,126.0 

24 

237.803 

206.610 

947.861 

1,154.471 

.060171 

16.620 

1,038.0 

26 

242.225 

211.089 

944.730 

1,155.819 

.064870 

15.420 

962.3 

28 

246.376 

215.293 

941.791 

1,157.084 

.069545 

14.380 

897.6 

30 

250.293 

219.261 

939.019 

1,158.280 

.074201 

13.480 

841.3 

32 

254.002 

223.021 

936.389 

1,159.410 

.078839 

12.680 

791.8 

34 

257.523 

226.594 

933.891 

1,160.485 

.083461 

11.980 

948.0 

36 

260.883 

230.001 

931.508 

1,161.509 

.088067 

11.360 

708.8 

38 

264.093 

233.261 

929.227 

1,162.488 

.092657 

10.790 

673.7 

40 

267.168 

236.386 

927.040 

1,163.426 

.097231 

10.280 

642.0 

42 

270.122 

239.389 

924.940 

1,164.329 

.101794 

9.826 

613.3 

44 

272.965 

242.275 

922.919 

1,165.194 

.106345 

9.403 

587.0 

46 

275.704 

245.061 

920.968 

1,166.029 

.110884 

9.018 

563.0 

48 

278.348 

247.752 

919.084 

1,166.836 

.115411 

8.665 

540.9 

50 

280.904 

250.355 

917.260 

1,167.615 

.119927 

8.338 

520.5 

52 

283.381 

252.875 

915.494 

1,168.369 

.124433 

8.037 

501.7 

54 

285.781 

255.321 

913.781 

1,169.102 

.128928 

7.756 

484.2 

56 

288.111 

257.695 

912.118 

1,169.813 

.133414 

7.496 

467.9 

58 

290.374 

260.002 

910.501 

1,170.503 

.137892 

7.252 

452.7 

60 

292.575 

262.248 

908.928 

1,171.176 

.142362 

7.024 

438.5 

62 

294.717 

264.433 

907.396 

1,171.829 

.146824 

6.811 

425.2 

64 

296.805 

266.566 

905.900 

1,172.466 

.151277 

6.610 

412.6 

66 

298.842 

268.644 

904.443 

1,173.087 

.155721 

6.422 

400.8 

68 

300.831 

270.674 

903.020 

1,173.694 

.160157 

6.244 

389.8 

70 

302.774 

272.657 

901.629 

1,174.286 

.164584 

6.076 

379.3 

72 

304.669 

274.597 

900.269 

1,174.866 

.169003 

5.917 

369.4 

74 

306.526 

276.493 

898.938 

1,175.431 

.173417 

5.767 

360.0 

76 

308.344 

278.350 

897.635 

1,175.985 

.177825 

5.624 

351.1 

78 

310.123 

280.170 

896.359 

1,176.529 

.182229 

5.488 

342.6 

80 

311.866 

281.952 

895.108 

1,177.060 

.186627 

5.358 

334.5 

82 

313.576 

283.701 

893.879 

1,177.580 

.191017 

5.235 

326.8 

84 

315.250 

285.414 

892.677 

1,178.091 

.195401 

5.118 

319.5 

86 

316.893 

287.096 

891.496 

1,178.592 

.199781 

5.006 

312.5 

88 

318.510 

288.750 

890.335 

1,179.085 

.204155 

4.898 

305.8 

STEAM  TABLES. 


31 


Table— ( Continued) . 


1 

2 

3 

4 

5 

6 

7 

8 

p 

t 

Q 

L 

H 

W 

V 

R 

90 

320.094 

290.373 

889.196 

1,179.569 

.208525 

4.796 

299.4 

92 

321.653 

291.970 

888.075 

1,180.045 

.212892 

4.697 

293.2 

94 

323.183 

293.539 

886.972 

1,180.511 

.217253 

4.603 

287.3 

96 

324.688 

295.083 

885.887 

1,180.970 

.221604 

4.513 

281.7 

98 

326.169 

296.601 

884.821 

1,181.422 

.225950 

4.426 

276.3 

100 

327.625 

298.093 

883.773 

1,181.866 

.230293 

4.342 

271.1 

105 

331.169 

301.731 

881.214 

1,182.945 

.241139 

4.147 

258.9 

110 

334.582 

305.242 

878.744 

1,183.986 

.251947 

3.969 

247.8 

115 

337.874 

308.621 

876.371 

1,184.992 

.262732 

3.806 

237:6 

120 

341.058 

311.885 

874.076 

1,185.961 

.273500 

3.656 

228.3 

125 

344.136 

315.051 

871.848 

1,186.899 

.284243 

3.518 

219.6 

130 

347.121 

318.121 

869.688 

1,187.809 

.294961 

3.390 

211.6 

135 

350.015 

321.105 

867.590 

1,188.695 

.305659 

3.272 

204.2 

140 

352.827 

324.003 

865.552 

1,189.555 

.316338 

3.161 

197.3 

145 

355.562 

326.823 

863.567 

1,190.390 

.326998 

3.058 

190.9 

150 

358.223 

329.566 

861.634 

1,191.200 

.337643 

2.962 

184.9 

160 

363.346 

334.850 

857.912 

1,192.762 

.358886 

2.786 

173.9 

170 

368.226 

339.892 

854.359 

1,194.251 

.380071 

2.631 

164.3 

180 

372.886 

344.708 

850.963 

1,195.671 

.401201 

2.493 

155.6 

190 

377.352 

349.329 

847.703 

1,197.032 

.422280 

2.368 

147.8 

200 

381.636 

353.766 

844.573 

1,198.339 

.443310 

2.256 

140.8 

210 

385.  / 59 

358.041 

841.556 

1,199.597 

.464295 

2.154 

134.5 

220 

389.736 

362.168 

838.642 

1,200.810 

.485237 

2.061 

128.7 

230 

393.575 

366.152 

835.828 

1,201.980 

.506139 

1.976 

123.3 

240 

397.285 

370.008 

833.103 

1,203.111 

.527003 

1.898 

118.5 

250 

400.883 

373.750 

830.459 

1,204.209 

.547831 

1.825 

114.0' 

260 

404.370 

377.377 

827.896 

1,205.273 

.568626 

1.759 

109.8 

270 

407.755 

380.905 

825.401 

1,206.306 

.589390 

1.697 

105.9 

280 

411.048 

384.337 

822.973 

1,207.310 

.610124 

1.639 

102.3 

290 

414.250 

387.677 

820.609 

1,208.286 

.630829 

1.585 

99.0’ 

300 

417.371 

390.933 

818.305 

1,209.238 

.651506 

1.535 

95.8 

32 


USEFUL  TABLES. 


LOGARITHMS. 


EXPONENTS. 

By  the  use  of  logarithms,  the  processes  of  multiplication, 
division,  involution,  and  evolution  are  greatly  shortened,  and 
some  operations  may  be  performed  that  would  be  impossible 
without  them.  Ordinary  logarithms  cannot  be  applied  to 
addition  and  subtraction. 

The  logarithm  of  a number  is  that  exponent  by  which  some 
fixed  number,  called  the  base,  must  be  affected  in  order  to 
equal  the  number.  Any  number  may  be  taken  as  the  base. 
Suppose  we  choose  4.  Then  the  logarithm  of  16  is  2,  because 
2 is  the  exponent  by  which  4 (the  base)  must  be  affected  in 
order  to  equal  16,  since  42  = 16.  In  this  case,  instead  of 
reading  42  as  4 square,  read  it  4 exponent  2.  With  the  same 
base,  the  logarithms  of  64  and  8 would  be  3 and  1.5,  respect- 
ively, since  43  = 64,  and  41-5  = 45  = 8.  In  these  cases,  as  in 
the  preceding,  read  43  and  41-6  as  4 exponent  3,  and  4 expo- 
nent 1.5,  respectively. 

Although  any  positive  number  except  1 can  be  used  as  a 
base  and  a table  of  logarithms  calculated,  but  two  numbers 
have  ever  been  employed.  For  all  arithmetical  operations 
(except  addition  and  subtraction)  the  logarithms  used  are 
called  the  Briggs,  or  common,  logarithms,  and  the  base  used 
is  10.  In  abstract  mathematical  analysis,  the  logarithms  used 
are  variously  called  hyperbolic,  Napierian , or  natural  loga- 
rithms, and  the  base  is  2.718281828+.  The  common  logarithm 
of  any  number  may  be  converted  into  a Napierian  logarithm 
by  multiplying  the  common  logarithm  by  2.30258509+,  which 
is  usually  expressed  as  2.3026,  and  sometimes  as  2.3.  Only 
the  common  system  of  logarithms  will  be  considered  here. 

Since  in  the  common  system  the  base  is  10,  it  follows  that, 
since  101  = 10, 102  = 100, 103  = 1,000,  etc.,  the  logarithm  (ex- 
ponent) of  10  is  1,  of  100  is  2,  of  1,000  is  3,  etc.  For  the  sake  of 
brevity  in  writing,  the  words  “ logarithm  of”  are  abbreviated 
to  “log.”  Thus,  instead  of  writing  logarithm  of  100  = 2, 
write  log  100  = 2.  When  speaking,  however,  the  words  for 
which  “ log”  stands  should  always  be  pronounced  in  full. 


LOGARITHMS. 


33 


From  the  above  it  will  be  seen  that,  when  the  base  is  10, 
since  10°  = 1,  the  exponent  0 = log  1; 

since  101  = 10,.  the  exponent  1 = log  10; 

since  102  = 100,  the  exponent  2 = log  100; 

since  103  = 1,000,  the  exponent  3 = log  1,000;  etc. 

Also, 

since  10-1  = = .1,  the  exponent  — 1 = log  .1; 

since  10~2  = = .01,  .the  exponent  — 2 = log  .01; 

since  10~3  = = *001,  the  exponent  — 3 = log  .001;  etc. 

From  this  it  will  be  seen  that  the  logarithms  of  exact 
powers  of  10  and  of  decimals  like  .1,  .01,  and  .001  are  the 
whole  numbers  1,  2,  3,  etc.  and  —1,  —2,  —3,  etc.,  respectively. 
Only  numbers  consisting  of  1 and  one  or  more  ciphers  have 
whole  numbers  for  logarithms. 

Now,  it  is  evident  that,  to  produce  a number  between  1 
and  10,  the  exponent  of  10  must  be  a fraction;  to  produce 
a number  between  10  and  100,  it  must  be  1 plus  a fraction; 
to  produce  a number  between  100  and  1,000,  it  must  be  2 plus 
a fraction;  etc.  Hence,  the  logarithm  of  any  number  between 
1 and  10  is  a fraction;  of  any  number  between  10  and  100, 

1 plus  a fraction;  of  any  number  between  100  and  1,000, 

2 plus  a fraction,  etc.  A logarithm,  therefore,  usually  con- 
sists of  two  parts:  a whole  number,  called  the  characteristic , 
and  a fraction,  called  the  mantissa.  The  mantissa  is  always 
expressed  as  a decimal.  For  example,  to  produce  20, 10  must 
have  an  exponent  of  approximately  1.30103,  or  101. 30103  = 20, 
very  nearly,  the  degree  of  exactness  depending  on  the  num- 
ber of  decimal  places  used.  Hence,  log  20  = 1.30103,  1 being 
the  characteristic,  and  .30103,  the  mantissa. 

Referring  to  the  second  part  of  the  preceding  table,  it  is 
clear  that  the  logarithms  of  all  numbers  less  than  1 are  nega- 
tive, the  logarithms  of  those  between  1 and  .1  being  —1  plus 
a fraction.  For,  since  log  .1  = —1,  the  logarithms  of  .2,  .3,  etc. 
(which  are  all  greater  than  .1,  but  less  than  1)  must  be 
greater  than  —1;  i.  e.,  they  must  equal  —1  plus  a fraction. 
For  the  same  reason,  to  produce  a number  between  .1  and 
.01,  the  logarithm  (exponent  of  10)  would  be  equal  to  —2 
plus  a fraction,  and  for  a number  between  .01  and  .001,  it 
would  be  equal  to  —3  plus  a fraction.  Hence,  the  logarithm 


34 


USEFUL  TABLES. 


of  any  number  between  1 and  .01  has  a negative  character- 
istic of  1 and  a positive  mantissa;  of  a number  between 
.1  and  .01,  a negative  characteristic  of  2 and  a positive 
mantissa;  of  a number  between  .01  and  .001,  a negative 
characteristic  of  3 and  a positive  mantissa;  of  a number 
between  .001  and  .0001,  a negative  characteristic  of  4 and  a 
positive  mantissa,  etc.  The  negative  characteristics  are  dis- 
tinguished from  the  positive  by  the  — sign  written  over  the  char- 
acteristic. Thus,  3 indicates  that  3 is  negative. 

It  must  be  remembered  that  in  all  cases  the  mantissa  is  posi- 
tive. Thus^the  logarithm  1.30103  means  +1  + .30103,  and  the 
logarithm  1.30103  means  —1  + .30103.  Were  the  minus  sign 
written  in  front  of  the  characteristic,  it  would  indicate  that 
the  entire  logarithm  was  negative.  Thus,  —1.30103  = —1 
-.30103. 

Rule  for  Characteristic.— Startingfrom  the  unit  figure,  count 
the  number  of  places  to  the  first  (left-hand)  digit  of  the  given 
number,  calling  unit’s  place  zero;  the  number  of  places  thus 
counted  will  be  the  required  characteristic.  If  the  first  digit 
lies  to  the  left  of  the  unit  figure,  the  characteristic  is  positive; 
if  to  the  right,  negative.  If  the  first  digit  of  the  number  is  the 
unit  figure,  the  characteristic  is  0.  Thus,  the  charactertisic  of 
the  logarithm  of  4,826  is  3,  since  the  first  digit,  4,  lies  in  the  3d 
place  to  the  left  of  the  unit  figure,  6.  The  characteristic  of 
the  logarithm  of  0.0000072  is  —6  or  6 , since  the  first  digit,  7, 
lies  in  the  6th  place  to  the  right  of  the  unit  figure.  The  char- 
acteristic of  the  logarithm  of  4.391  is  0,  since  4 is  both  the  first 
digit  of  the  number  and  also  the  unit  figure. 


TO  FIND  THE  LOGARITHM  OF  A NUMBER. 

To  aid  in  obtaining  the  mantissas  of  logarithms,  tables  of 
logarithms  have  been  calculated,  some  of  which  are  very 
elaborate  and  convenient.  In  the  Table  of  Logarithms,  the 
mantissas  of  the  logarithms  of  numbers  from  1 to  9,999  are 
given  to  five  places  of  decimals.  The  mantissas  of  logarithms 
of  larger  numbers  can  be  found  by  interpolation.  The  table 
contains  the  mantissas  only;  the  characteristics  may  be  easily 
found  by  the  preceding  rule. 


LOGARITHMS. 


35 


The  table  depends  on  the  principle,  which  will  be 
explained  later,  that  all  numbers  having  the  same  figures 
in  the  same  order  have  the  same  mantissa,  without  regard  to 
the  position  of  the  decimal  point,  which  affects  the  charac- 
teristic only.  To  illustrate,  if  log  206  = 2.31387,  then, 
log  20.6  = 1.31387;  log  .206  = 1.31387; 

log  2.06  = .31387;  log  .0206  = 2.31387;  etc. 

To  find  the  logarithm  of  a number  not*  having  more  than 
four  figures: 

Rule. — Find  the  first  three  significant  figures  of  the  number 
whose  logarithm  is  desired , in  the  left-hand  column;  find  the 
fourth  figure  in  the  column  at  the  top  (or  bottom)  of  the  page;  and 
in  the  column  under  (or  above)  this  figure,  and  opposite  the  first 
three  figures  previously  found,  will  be  the  mantissa  or  decimal 
pari  of  the  logarithm.  The  characteristic  being  found  as  pre- 
viously described,  write  it  at  the  left  of  the  mantissa,  and  the 
resulting  expression  will  be  the  logarithm  of  the  required  number. 

Example.— Find  from  the  table  the  logarithm  (a)  of  476; 
(b)  of  25.47;  (c)  of  1.073;  (d)  of  .06313. 

Solution.— (a)  In  order  to  economize  spaoe  and  make 
the  labor  of  finding  the  logarithms  easier,  the  first  two  figures 
of  the  mantissa  are  given  only  in  the  column  headed  0.  The 
last  three  figures  of  the  mantissa,  opposite  476  in  the  column 
headed  N (N  stands  for  number),  are  761,  found  in  the 
column  headed  0;  glancing  upwards,  we  find  the  first  two 
figures  of  the  mantissa,  viz.,  67.  The  characteristic  is  2; 
hence,  log  476  = 2.67761. 

Note.— Since  all  numbers  in  the  table  are  decimal  frac- 
tions, the  decimal  point  is  omitted  throughout;  this  is  cus- 
tomary in  all  tables  of  logarithms. 

(b)  To  find  the  logarithm  of  25.47,  we  find  the  first  three 
figures,  254,  in  the  column  headed  N,  and  on  the  same  hori- 
zontal line,  under  the  column  headed  7 (the  fourth  figure  of 
the  given  number),  will  be  found  the  last  three  figures  of 
the  mantissa,  viz.,  603.  The  first  two  figures  are  evidently  40, 
and  the  characteristic  is  1;  hence,  log  25.47  = 1.40603. 

(c)  For  1.073;  in  the  column  headed  3,  opposite  107  in  the 
column  headed  N,  the  last  three  figures  of  the  mantissa  are 
found,  in  the  usual  manner,  to  be  060.  It  will  be  noticed 


36 


USEFUL  TABLES. 


that  these  figures  are  printed  *060,  the  star  meaning  that 
instead  of  glancing  upwards  in  the  column  headed  0,  and 
taking  02  for  the  first  two  figures,  we  must  glance  downwards 
and  take  the  two  figures  opposite  the  number  108,  in  the 
left-hand  column,  i.  e.,  03.  The  characteristic  being  0,  log 
1.073  = 0.03060$  or,  more  simply,  .03060. 

(d)  For  .06313;  the  last  three  figures  of  the  mantissa  are 
found  opposite  631,  in  column  headed  3,  to  be  024.  In  this 
case,  the  first  two  figures  .occur  in  the  same  row,  and  are  80. 
Since  the  characteristic  is  2,  log  .06313  = 2.80024. 

If  the  original  number  contains  but  one  digit  (a  cipher  is 
not  a digit),  annex  mentally  two  ciphers  to  the  right  of  the 
digit;  if  the  number  contains  but  two  digits  (with  no  ciphers 
between,  as  in  4,008),  annex  mentally  one  cipher  on  the 
right  before  seeking  the  mantissa.  Thus,  if  the  logarithm  ot 
7 is  wanted,  seek  the  mantissa  for  700,  which  is  .84510;  or,  if 
the  logarithm  of  48  is  wanted,  seek  the  mantissa  for  480, 
which  is  .68124.  Or,  find  the  mantissas  of  logarithms  of  num- 
bers between  0 and  100,  on  the  first  page  of  the  tables. 

The  process  of  finding  the  logarithm  of  a number  from  the 
table  is  technically  called  taking  out  the  logarithm. 

To  take  out  the  logarithm  of  a number  consisting  of  more 
than  four  figures,  it  is  inexpedient  to  use  more  than  five 
figures  of  the  number  when  using  five-place  logarithms  (the 
logarithms  given  in  the  accompanying  table  are  five-place). 
Hence,  if  the  number  consists  of  more  than  five  figures  and 
the  sixth  figure  is  less  than  5,  replace  all  figures  after  the  fifth 
with  ciphers;  if  the  sixth  figure  is  5 or  greater,  increase  the 
fifth  figure  by  1 and  replace  the  remaining  figures  with 
ciphers.  Thus,  if  the  number  is  31,415,926,  find  the  logarithm 
of  31,416,000;  if  31,415,426,  find  the  logarithm  of  31,415,000. 

Example.— Find  log  31,416. 

Solution.— Find  the  mantissa  of  the  logarithm  of  the  first 
four  figures,  as  explained  above.  This  is,  in  the  present  case, 
.49707.  Now,  subtract  the  number  in  the  column  headed  1, 
opposite  314  (the  first  three  figures  of  the  given  number),  from 
the  next  greater  consecutive  number,  in  this  case  721,  in  the 
column  headed  2.  721  — 707  = 14;  this  number  is  called  the 
difference.  At  the  extreme  right  of  the  page  will  be  found  a 


LOGARITHMS. 


37 


secondary  table  headed  P.  P.,  and  at  the  top  of  one  of  these 
columns,  in  this  table,  in  bold-face  type,  will  be  found  the 
difference.  It  will  be  noticed  that  each  column  is  divided 
into  two  parts  by  a vertical  line,  and  that  the  figures  on  the 
left  of  this  line  run  in  sequence  from  1 to  9.  Considering  the 
difference  column  headed  14,  we  see  opposite  the  number  6 
(6  is  the  last  or  fifth  figure  of  the  number  whose  logarithm 
we  are  taking  out)  the  number  8.4,  and  we  add  this  number 
to  the  mantissa  found  above,  disregarding  the  decimal  point 
in  the  mantissa,  obtaining  49,707  + 8.4  = 49,715.4.  Now,  since 
4 is  less  than  5,  wre  reject  it,  and  obtain  for  our  complete 
mantissa  .49715.  Since  the  characteristic  of  the  logarithm  of 
31,416  is  4,  log  31,416  = 4.49715. 

Example.— Find  log  380.93. 

Solution.— Proceeding  in  exactly  the  same  manner  as 
above,  the  mantissa  for  3,809  is  58,081  (the  star  directs  us  to  take 
58  instead  of  57  for  the  first  two  figures);  the  next  greater 
mantissa  is  58,092,  found  in  the  column  headed  0,  opposite  381 
in  column  headed  N.  The  difference  is  092  — 081  = 11.  Look- 
ing in  the  section  headed  P.  P.  for  column  headed  11,  we  find 
opposite  3,  3.3;  neglecting  the  .3,  since  it  is  less  than  5,  3 is  the 
amount  to  be  added  to  the  mantissa  of  the  logarithm  of  3,809 
to  form  the  logarithm  of  38,093.  Hence,  58,081  + 3 = 58,084, 
and  since  the  characteristic  is  2,  log  380.93  = 2.58084. 

Example— Find  log  1,296,728. 

Solution.— Since  this  number  consists  of  more  than  five 
figures  and  the  sixth  figure  is  less  than  5,  we  find  the  loga- 
rithm of  1,296,700  and  call  it  the  logarithm  of  1,296,728.  The 
mantissa  of  log  1,296  is  found  to  be  11,261.  The  difference  is 
294  — 261  = 33.  Looking  in  the  P.  P.  section  for  column 
headed  33,  we  find  opposite  7,  on  the  extreme  left,  23.1;  neg- 
lecting the  .1,  the  amount  to  be  added  to  the  above  mantissa 
is  23.  Hence,  the  mantissa  of  log  1,296,728  = 11,261  + 23  = 
11,284;  since  the  characteristic  is  6,  log  1,296,728  = 6.11284. 

Example.— Find  log  89.126. 

Solution.— Log  89.12  = 1.94998.  Difference  between  this 
and  log  80.13  = 1.95002  — 1.94998  = 4.  The  P.  P.  (propor- 
tional part)  for  the  fifth  figure  of  the  number  6 is  2.4,  or  2. 

Hence,  log  89.126  = 1.94998  + .00002  = 1.95000. 


38 


USEFUL  TABLES. 


Example.— Find  log  .096725. 

Solution.—  Log  .09672  = 2.98552.  Difference  = 4. 

P.  P.  for  5 = 2 

Hence,  log  .096725  = 2.98554. 

To  find  the  logarithm  of  a number  consisting  of  five  or 
more  figures: 

Rule. — I.  If  the  number  consists  of  more  than  five  figures  and 
the  sixth  figure  is  5 or  greater , increase  the  fifth  figure  by  1 and 
write  ciphers  in  place  of  the  sixth  and  remaining  figures. 

II.  Find  the  mantissa  corresponding  to  the  logarithm  of  the 
first  four  figures , and  substract  this  mantissa  from  the  next  greater 
mantissa  in  the  table;  the  remainder  is  the  difference. 

III.  Find  in  the  secondary  table  headed  P.  P.  a column 
headed  by  the  same  number  as  that  just  found  for  the  difference , 
and  in  this  column , opposite  the  number  corresponding  to  the  fifth 
figure  {or  fifth  figure  increased  by  1)  of  the  given  number  {this 
figure  is  always  situated  at  the  left  of  the  dividing  line  of  the 
column ),  will  be  found  the  P.  P.  {proportional  part)  for  that 
number.  The  P.  P.  thus  found  is  to  be  added  to  the  mantissa 
found  in  II,  as  in  the  preceding  examples , and  the  result  is  the 
mantissa  of  the  logarithm  of  the  given  number , as  nearly  as  may 
be  found  with  five-place  tables. 


TO  FIND  A NUMBER  WHOSE  LOGARITHM  IS  GIVEN. 

Rule.  —I.  Consider  the  mantissa  first.  Glance  along  the  differ- 
ent columns  of  the  table  which  are  headed  0,  until  the  first  two 
figures  of  the  mantissa  are  found.  Then,  glance  down  the  same 
column  until  the  third  figure  is  found  {or  1 less  than  the  third 
figure) . Having  found  the  first  three  figures,  glance  to  the  right 
along  the  row  in  which  they  are  situated  until  the  last  three  figures 
of  the  mantissa  are  found.  Then,  the  number  that  heads  the 
column  in  which  the  last  three  figures  of  the  mantissa  are  found 
is  the  fourth  figure  of  the  required  number,  and  the  first  three 
figures  lie  in  the  column  headed  N,  and  in  the  same  row  in  which 
lie  the  last  three  figures  of  the  mantissa. 

II.  If  the  mantissa  cannot  be  found  in  the  table,  find  the 
mantissa  that  is  nearest  to,  but  less  than,  the  given  mantissa,  and 
which  call  the  next  less  mantissa.  Subtract  the  next  less  mantissa 


LOGARITHMS. 


39 


from  the  next  greater  mantissa  in  the  table  to  obtain  the  difference. 
Also,  subtract  the  next  less  mantissa  from  the  mantissa  of  the 
given  logarithm,  and  call  the  remainder  the  P.  P.  Looking  in 
the  secondary  table  headed  P.  P.  for  the  column  headed  by  the 
difference  just  found,  find  the  number  opposite  the  P.  P.  just 
found  (or  the  P.  P.  corresponding  most  nearly  to  that  just  found); 
this  number  is  the  fifth  figure  of  the  required  number ; the  fourth 
figure  will  be  found  at  the  top  of  the  column  containing  the  next 
less  mantissa,  and  the  first  three  figures  in  the  column  headed  N 
and  in  the  same  row  that  contains  the  next  less  mantissa. 

III.  Having  found  the  figures  of  the  number  as  above 
directed,  locate  the  decimal  point  by  the  rules  for  the  character- 
istic, annexing  ciphers  to  bring  the  number  up  to  the  required 
number  of  figures  if  the  characteristic  is  greater  than  4. 

Example.— Find  the  number  whose  logarithm  is  3.56867. 

Solution.— The  first  two  figures  of  the  mantissa  are  56; 
glancing  down  the  column,  we  find  the  third  figure,  8 (in  con- 
nection with  820),  opposite  370  in  the  N column.  Glancing 
to  the  right  along  the  row  containing  820,  the  last  three 
figures  of  the  mantissa,  867,  are  found  in  the  column  headed 
4;  hence,  the  fourth  figure  of  the  required  number  is  4,  and 
the  first  three  figures  are  370,  making  the  figures  of  the 
required  number  3,704.  Since  the  characteristic  is  3,  there 
are  three  figures  to  the  left  of  the  unit  figure,  and  the  number 
whose  logarithm  is  3.56867  is  3,704. 

Example.— Find  the  number  whose  logarithm  is  3.56871. 

Solution.— The  mantissa  is  not  found  in  the  table.  The 
next  less  mantissa  is  56,867;  the  difference  between  this  and 
the  next  greater  mantissa  is  879  — 867  = 12,  and  the  P.  P.  is 
56,871  — 56,867  = 4.  Looking  in  the  P.  P.  section  for  the 
column  headed  12*  we  do  not  find  4,  but  we  do  find  3.6  and 
4.8.  Since  3.6  is  nearer  4 than  4.8,  we  take  the  number 
opposite  3.6  for  the  fifth  figure  of  the  required  number;  this 
is  3.  Hence,  the  fourth  figure  is  4;  the  first  three  figures 
370,  and  the  figures  of  the  number  are  37,043.  The  charac- 
teristic being  3,  the  number  is  3,704.3. 

Example.— Find  the  number  whose  logarithm  is  5.95424. 

Solution. — The  mantissa  is  found  in  the  column  headed  0, 
opposite  900  in  the  column  headed  N.  Hence,  the  fourth 


40 


USEFUL  TABLES. 


figure  is  0,  and  the  number  is  900,000,  the  characteristic 
being  5.  Had  the  logarithm  been  5.95424,  the  number  would 
have  been  .00009. 

Example.— Find  the  number  whose  logarithm  is  .93036. 

Solution.— The  first  three  figures  of  the  mantissa,  930, 
are  found  in  the  0 column,  opposite  852  in  the  N column; 
but  since  the  last  two  figures  of  all  the  mantissas  in  this  row 
are  greater  than  36,  we  must  seek  the  next  less  mantissa  in 
the  preceding  row.  We  find  it  to  be  93,034  (the  star  directing 
us  to  use  93  instead  of  92  for  the  first  two  figures),  in  the 
column  headed  8.  The  difference  for  this  case  is  039  — 034 
= 5,  and  the  P.  P.  is  036  — 034  = 2.  Looking  in  the  P.  P. 
section  for  the  column  headed  5,  we  find  the  P.  P.,  2,  opposite  4. 
Hence,  the  fifth  figure  is  4;  the  fourth  figure  is  8;  the  first 
three  figures  851,  and  the  number  is  8.5184,  the  characteristic 
being  0. 

Example.— Find  the  number  whose  logarithm  is  2.05753. 

Solution.— The  next  less  mantissa  is  found  in  column 
headed  1,  opposite  114  in  the  N column;  hence,  the  first 
four  figures  are  1,141.  The  difference  for  this  case  is  767  — 729 
= 38,  and  the  P.  P.  is  753  — 729  = 24.  Looking  in  the  P.  P. 
section  for  the  column  headed  38,  we  find  that  24  falls 
between  22.8  and  26.6.  The  difference  between  24  and  22.8 
is  1.2,  and  between  24  and  26.6  is  2.6;  hence,  24  is  nearer  22.8 
than  it  is  to  26.6,  and  6,  opposite  22.8,  is  the  fifth  figure  of  the 
number.  Hence,  the  number  whose  logarithm  is  2.05753 
is  .011416. 

In  order  to  calculate  by  means  of  logarithms,  a table  is 
absolutely  necessary.  Hence,  for  this  reason,  we  do  not 
explain  the  method  of  calculating  a logarithm.  The  work 
involved  in  calculating  even  a single  logarithm  is  very  great, 
and  no  method  has  yet  been  demonstrated,  of  which  we  are 
aware,  by  which  the  logarithm  of  a number  like  121  can  be 
calculated  directly.  Moreover,  even  if  the  logarithm  could 
be  readily  obtained,  it  would  be  useless  without  a complete 
table,  such  as  that  which  is  here  given,  for  the  reason  that 
«,fter  having  used  it,  say  to  extract  a root,  the  number 
corresponding  to  the  logarithm  of  the  result  could  not 
be  found. 


LOGARITHMS. 


41 


MULTIPLICATION  BY  LOGARITHMS. 

The  principle  upon  which  the  process  is  based  may  be 
illustrated  as  follows:  Let  X and  Y represent  two  numbers 
whose  logarithms  are  x and  y.  To  find  the  logarithm  of  their 
product,  we  have,  from  the  definition  of  a logarithm, 

10*  = X,  (1) 
and  10y  = Y.  (2) 

Since  both  members  of  (1)  may  be  multiplied  by  the  same 
quantity  without  destroying  the  equality,  they  evidently 
may  be  multiplied  by  equal  quantities  like  10y  and  Y.  Hence, 
multiplying  (1)  by  (2),  member  by  member, 

10*X10V  = 10v+y  = XY, 

or,  by  the  definition  of  a logarithm,  x + y = log  X Y.  But 
X Y is  the  product  of  X and  Y,  and  x + y is  the  sum  of  their 
logarithms;  from  which  it  follows  that  the  sum  of  the  loga- 
rithms of  two  numbers  is  equal  to  the  logarithm  of  their 
product.  Hence, 

To  multiply  two  or  more  numbers  by  using  logarithms: 

Rule. — Add  the  logarithms  of  the  several  numbers , and  the  sum 
will  be  the  logarithm  of  the  product.  Find  the  number  corre- 
sponding to  this  logarithm,  and  the  result  will  be  the  number 
sought. 

Example.— Multiply  4.38,  5.217,  and  83  together. 

Solution.—  Log  4.38  = .64147 
Log  5.217  = .71742 
Log  83  = 1.91908 

Adding,  3.27797  = log  (4.38  X 5.217  X 83). 

Number  corresponding  to  3.27797  = 1,896.6.  Hence,  4.38 
X 5.217  X 83  = 1,896.6,  nearly.  By  actual  multiplication,  the 
productls  1,896.5818,  showing  that  the  result  obtained  by  using 
logarithms  was  correct  to  five  figures. 

When  adding  logarithms,  their  algebraic  sum  is  always 
to  be  found.  Hence,  if  some  of  their  numbers  multiplied 
together  are  wholly  decimal,  the  algebraic  sum  of  the  char- 
acteristics will  be  the  characteristic  of  the  product.  It  must 
be  remembered  that  the  mantissas  are  always  positive. 

Example.— Multiply  49.82,  .00243,  17,  and  .97  together. 


42 


USEFUL  TABLES. 


Solution  — 

Log  49.82  = 1.69740 
Log  .00243  = 3.38561 
Log  17  = 1.23045 

Log  .97  = 1.98677 

Adding,  0.30023  = log  (49.82  X .00243  X 17  X .97) . 

Number  corresponding  to  0.30023  = 1.9963.  Hence,  49.82 
X .00243  X 17  X .97  = 1.9963. 

In  this  case  the  sum  of  the  mantissas  was  2.30023.  The 
integral  2 added  to  the  positive  characteristics  makes  their 
sum  = 2 + 1 + 1 = 4;  sum  of  negative  characteristics  = 3 
+ 1 = 4,  whence  4 + (— 4)  = 0.  If,  instead  of  17,  the  number 
had  been  .17  in  the  above  example,  the  logarithm  of  .17  would 
have  been  1.23045,  and  the  sum  of  the  logarithms  would  have 
been  2.30023;  the  product  would  then  have  been  .019963. 

It  can  now  be  shown  why  all  numbers  with  figures  in  the 
same  order  have  the  same  mantissa,  without  regard  to  the 
decimal  point.  Thus,  suppose  it  were  known  that  log  2.06 
= .31387.  Then,  log  20.6  = log  (2.06  X 10)  = log  2.06  + log  10 
= .31387  + 1 = 1.31387.  And  so  it  might  be  proved  with  the 
decimal  point  in  any  other  position. 


DIVISION  BY  LOGARITHMS. 

As  before,  let  X and  Y represent  two  numbers  whose  loga- 
rithms are  x and  y.  To  find  the  logarithm  of  their  quotient, 
we  have,  from  the  definition  of  a logarithm, 

10*  = X,  (1) 
and  10y  = Y.  (2) 

Dividing  (1)  by  (2),  ltf~y  =»  or,  by  the  definition  of  a 
X X 

logarithm,  x — y = log  — . But  -y  is  the  quotient  of  X-r-  Y, 

and  x — y is  the  difference  of  their  logarithms,  from  which  it 
follows  that  the  difference  between  the  logarithms  of  two  numbers 
is  equal  to  the  logarithm  of  their  quotient.  Hence,  to  divide 
one  number  by  another  by  means  of  logarithms : 

Rule. — Subtract  the  logarithm  of  the  divisor  from  the  logarithm 
of  the  dividend , and  the  result  will  be  the  logarithm  of  the  quotient. 


LOGARITHMS. 


43 


Example.— Divide  6,784.2  by  27.42. 

Solution.—  Log  6,784.2  = 3.83150 
Log  27.42  = 1.43807 

difference  = 2.39343  = log  (6,784.2-=- 27.42). 

Number  corresponding  to  2.39343  = 247.42.  Hence,  6,784.2 
-4-  27.42  = 247.42. 

When  subtracting  logarithms,  their  algebraic  difference  is 
to  be  found.  The  operation  may  sometimes  be  confusing, 
because  the  mantissa  is  always  positive,  and  the  character- 
istic may  be  either  positive  or  negative.  When  the  logarithm 
to  be  subtracted  is  greater  than  the  logarithm  from  which  it  is  to 
be  taken,  or  when  negative  characteristics  appear,  subtract  the 
mantissa  first , and  then  the  characteristic,  by  changing  its  sign 
and  adding. 

Example.— Divide  274.2  by  6,784.2. 

Solution.—  Log  274.2  = 2.43807 
Log  6,784.2  = 3.83150 

2.60657 

First  subtracting  the  mantissa  .83150  gives  .60657  for  the 
mantissa  of  the  quotient.  In  subtracting,  1 had  to  be  taken 
from  the  characteristic  of  the  minuend,  leaving  a charac- 
teristic of  1.  Subtract  the  characteristic  3 from  this,  by 
changing  its  sign  and  adding  1 — 3 = 2,  the  characteristic  of 
the  quotient.  Number  corresponding  to  2.60657  = .040418. 
Hence,  274.2  -4-  6,784.2  = .040418. 

Example.— Divide  .067842  by  .002742. 

Solution.—  Log  .067842  = ^2.83150 
Log  .002742  = 3.43807 

difference  = 1.39343 

Since  .83150  — .43807  = .39343  and  — 2 + 3 = 1,  number  cor- 
responding to  1.39343  = 24.742.  Hence,  .067842  -4-  .002742  = 
24.742. 

The  only  case  that  is  likely  to  cause  trouble  in  subtract- 
ing is  that  in  which  the  logarithm  of  the  minuend  has  a nega- 
tive characteristic,  or  none  at  all,  and  a mantissa  less  than 
the  mantissa  of  the  subtrahend.  For  example,  let  it  be  re- 
quired to  subtract  the  logarithm  3.74036  from  the  logarithm 


44 


USEFUL  TABLES. 


3.55145.  The  logarithm  3.55145  is  equivalent  to—  3 + .55145. 
Now,  if  we  add  both  +1  and  —1  to  this  logarithm,  it  will  not 
change  its  value.  Hejice,  3.55145  = —3  — 1 + 1 + .55145  = 4 
+ 1.55145.  Therefore,  3.55145  — 3.74036  = 

4 + 1.55145 
3 + .74036 

difference  — 7 + .81109  = 7.81109. 

Had  the  characteristic  of  the  above  logarithm  been  0 
instead  of  3,  the  process  would  have  been  exactly  the  same. 
Thus,  .55145  = 1 +•  1.55145^  hence, 

1 + 1.55145 
3+  .74036 

difference  = 4 + .81109  = 4.81109. 

Example.— Divide  .02742  by_67.842. 

Solution.—  Log  .02742  = 2.43807  = 3 + 1.43807 
Log  67.842  = 1.83150  = 1 + .83150 

difference  = 4 + .60657  = 4.60657. 
Number  corresponding  to  4.60657  = .00040417.  Hence, 
.02742  -f-  67.842  = .00040417. 

Example. — What  is  the  reciprocal  of  3.1416? 

Solution.— Reciprocal  of  3.1416  = 0 * and  log  — ^ 

o.141d  0.1410 

= log  1 - log  3.1416  = 0 — _.49715.  Since  0 = -1  + 1, 

1 + 1.00000 
.49715 

difference  = 1 + ^50285  = 1.50285. 

Number  wThose  logarithm  is  1.50285  = .31831. 


INVOLUTION  BY  LOGARITHMS. 

If  X represents  a number  whose  logarithm  is  x , we  have, 
from  the  definition  of  a logarithm, 

10*  = X. 

Raising  both  numbers  to  some  power,  as  the  nth,  the 
equation  becomes 

l0*n  _ xn 

But  Xn  is  the  required  power  of  X,  and  xn  is  its  logarithm, 
from  which  it  follows  that  the  logarithm  of  a number 


LOGARITHMS. 


45 


multiplied  by  the  exponent  of  the  power  to  which  it  is  raised 
is  equal  to  the  logarithm  of  the  power.  Hence,  to  raise  a 
number  to  any  power  by  the  use  of  logarithms: 

Rule.—  Multiply  the  logarithm  of  the  number  by  the  exponent 
that  denotes  the  power  to  which  the  number  is  to  be  raised , and  the 
result  will  be  the  logarithm  of  the  required  power. 

Example.— What  is  (a)  the  square  of  7.92?  (6)  the  cube 
of  94.7?  (c)  the  1.6  power  of  512,  that  is,  the  value  of  5121*6? 

Solution.— (a)  Log  7.92  = .89873;  exponent  of  power  = 2. 
Hence,  .89873  X 2 = 1.79746  = log  7.922.  Number  correspond- 
ing to  1.79746  = 62.727.  Hence,  7.922  = 62.727,  nearly. 

(6)  Log  94.7  = 1.97635;  1.97635  X 3 = 5.92905  = log  94.73. 
Number  corresponding  to  5.92905  = 849,280,  nearly.  Hence, 
$4.7*  = 849,280,  nearly. 

(c)  Log  512!-6  = 1.6  X log  512  = 1.6  X 2.70927  = 4.334832, 
or  4.33483  (when  using  five-place  logarithms)  = log  21,619. 
Hence,  5121*6  = 21,619  nearly. 

If  the  number  is  wholly  decimal,  so  that  the  characteristic 
is  negative,  multiply  the  two  parts  of  the  logarithm  separately  by 
the  exponent  of  the  number.  If , after  multiplying  the  mantissa , 
the  product  has  a characteristic , add  it , algebraically , to  the  neg- 
ative characteristic  multiplied  by  the  exponent , and  the  result  will 
be  the  negative  characteristic  of  the  required  power. 

Example. — Raise  .0751  to  the  fourth  power. 

Solution.— Log  .07514  = 4 X log  .0751  = 4 X 2.87564.  Mul- 
tiplying the  parts  separately^  4X2  = 8 and  4 X .87564 
= 3.50256.  Adding  the  3 and  8,  3 + (—  8)  = — 5;  therefore, 
log  .07514  = 5.50256.  Number  corresponding  to  this  = 
.00003181.  Hence,  .0751*  = .00003181. 

A decimal  may  be  raised  to  a power  whose  exponent  con- 
tains a decimal  as  follows: 

Example.— Raise  .8  to  the  1.21  power. 

Solution.— Log  .81-21  = 1.21  X 1.90309.  There  are  several 
ways  of  performing  the  multiplication. 

First  Method.— Adding  the  characteristic  and  mantissa 
algebraically,  the  result  is  —.09691.  Multiplying  this  by  1.21 
gives  —.1172611,  or  —.11726,  when  using  five-place  logarithms. 
To  obtain  a positive  mantissa,  add  +1  and  —1;  whence, 
log  .8b2i  = — ! + 1 — .11726  = 1.88274. 


46 


USEFUL  TABLES. 


Second  Method.— Multiplying  the  characteristic  and  man- 
tissa separately  gives  —1.21  + 1.09274.  Adding  characteristic 
and  mantissa  algebraically,  gives  —.11726;  then,  adding  +1 
and  -1,  log  .81-21  = 1.88274. 

Third  Method—  Multiplying  the  characteristic  and  man- 
tissa separately  gives  —1.21  + 1.09274.  Adding  the  decimal 
part  of  the  characteristic  to  the  mantissa  gives  —1  + (—.21 
-i- 1.09274)  = 1.88274  = log  .81-*1.  The  number  corresponding 
to  the  logarithm  1.88274  = .76338. 

Any  one  of  the  above  three  methods  may  be  used,  but  we 
recommend  the  first  or  the  third.  The  third  is  the  most 
elegant  and  saves  figures,  but  requires  the  exercise  of  more 
caution  than  the  first  method  does.  Below  will  be  found  the 
entire  work  of  multiplication  for  both  .81-21  and  .8-21. 


1.90309 

1.21 


1.90309 

.21 


90309 

180618 

90309 


1.0927389 

—1.21 


90309 

180618 


-F 1.1896489 
—1  — .21 

1.9796489,  or  1.97965. 


1.8827389,  or  1.88274. 

In  the  second  case,  the  negative  decimal  obtained  by 
multiplying  —1  and  .21  was  greater  than  the  positive  decimal 
obtained  by  multiplying  .90309  and  .21;  hence,  +1  and  — 1 
were  added,  as  shown.  


EVOLUTION  BY  LOGARITHMS. 

If  X represents  a number  whose  logarithm  is  x,  we  have, 
from  the  definition  of  a logarithm, 

10*  = X. 

Extracting  some  root  of  both  members,  as  the  nth,  the 
equation  becomes 

10”  = i/x 

But  \/  AXs  the  required  root  of  X,  and  ^ is  its  logarithm, 
from  which  it  follows  that  the  logarithm  of  a number  divided 


LOGARITHMS. 


47 


by  the  index  of  the  root  to  be  extracted  is  equal  to  the 
logarithm  of  the  root.  Hence,  to  extract  any  root  of  a number 
by  means  of  logarithms: 

Rule. — Divide  the  logarithm  of  the  number  by  the  index  of  the 
root;  the  result  will  be  the  logarithm  of  the  root. 

Example.— Extract  (a)  the  square  root  of  77,851;  ( b ) the 
cube  root  of  698,970;  (c)  the  2.4  root  of  8,964,300. 

Solution— (a)  Log  77,851  = 4.89127;  the  index  of  the  root 
is  2;  hence,  log  \/  77,851  = 4.89127  -r-  2 = 2.44564;  number 
corresponding  to  this  = 279.02.  Hence,  i/  77,851  = 279.02, 
nearly. 

( b ) Log  #" 698,970  = 5.84446^-  3 = 1.94815  = log  88.746;  or, 
#/ 698,970  = 88.747,  nearly. 

(c)  Log  8,964,300  = 6.95251  --  2.4  = 2.89688  = log  788.64; 
or,  2 y/  8,964,300  = 788.64,  nearly. 

If  it  is  required  to  extract  a root  of  a number  wholly  deci- 
mal, and  the  negative  characteristic  will  not  exactly  contain 
the  index  of  the  root,  without  a remainder,  proceed  as  follows: 

Separate  the  two  parts  of  the  logarithm;  add  as  many  units  ( or 
parts  of  a unit)  to  the  negative  characteristic  as  will  make  it 
exactly  contain  the  index  of  the  root.  Add  the  same  number  to 
the  mantissa,  and  divide  both  parts  by  the  index.  The  result 
will  be  the  characteristic  and  mantissa  of  the  root. 

Example.— Extract  the  cube  root  of  .0003181. 

Solution.— Log  #/  .0003181  _ log  .0003181  _ 4.50256 
3 3 * 

(4  + 2 = 6)_+  (2  + .50256  = 2.50256). 

(6-4-3  = 2)  + (2.50256  --  3 = .83419); 
or,  log  #^  .0003181  = 2.83419  = log  .068263. 

Hence,  #".0003181  = .068263. 

Example.— Find  the  value  of  1‘v/. 0003181. 

0 T 1.41/—--  - log  .0003181  4.50256 

Solution.— Log  y .0003181  = — = — -. 

If  —.23  be  added  to  the  characteristic,  it  will  contain  1.41 
exactly  3 times.  Hence, 

[-4  + (—  .23)  = —4.23]  + (.23  + .50256  = .73256). 

( — 4.23  -f- 1.41  = 3)  + (.73256 -f- 1.41  = .51955); 
or,  log  x' v/. 0003181  - 3.51955  = log  .0033079. 

Hence,  .0003181  = .0033079. 


48 


USEFUL  TABLES. 


Example. — Solve  this  expression  by  logarithms: 
497  X .0181  X 762 
3, 300 X. 6517  ~~  ‘ 

Solution.—  Log  497  = 2.69636 
Log  .0181  = 2.25768 

Log  762  = 2.88195 

Log  product  = 3.83599 

Log  3,300  = 3.51851 

Log  .6517  = 1.81405 


Log  product  ==  3.33256 


Hence, 


3.83599  - 3.33256  = .50343  = log  3.1874. 
497  X. 0181X762 
3,300  X .6517 


„ „ , 3/  504,203X507  , . 

Example.— Solve  7 by  logarithms. 

Solution.—  Log  504,203  = 5.70260 
Log  507  = 2.70501 


Log  product  = 8.40761 
Log  1.75  = .24304 
Log  71.4  = 1.85370 
Log  87  = 1.93952 

Log  product  = 4.03626 


8.40761  - 4.03626 


= 1.45712  = log  28.65. 


Hence, 


% 


504,203  X 507 


= 28.65. 


1.75  X 71.4  X 87 

Logarithms  can  often  be  applied  to  the  solution  of 
equations. 

Example.— Solve  the  equation  2.43s5  = v/.0648. 
Solution.—  2.43s5  = v/.0648. 

■^^648 


Dividing  by  2.43, 


s5  = • 


2.43 


Taking  the  logarithm  of  both  numbers, 

5 X log  s = lQg  ^Q6f 8 — log  2.43; 


LOGARITHMS. 


49 


5 log  x = 


2.81158 

6 


.38561 


= Jl.  80193  - .38561 
= 1.41632. 

Dividing  by  5,  log  x = 1.88326; 
whence,  x = .7643. 

Example.— Solve  the  equation  4.5*  = 8. 

Solution.— Taking  the  logarithms  of  both  numbers, 
x log  4.5  = log  8, 
log  8 _ .90309 
.65321* 


whence, 


~~  log  4.5 

Taking  logarithms  again, 

log  x = log  .90309  - log  .65321  = 1.95573  — 1.81505 
= .14068;  whence,  x = 1.3825. 

Remark. — Logarithms  are  particularly  useful  in  those 
cases  when  the  unknown  quantity  is  an  exponent,  as  in  the 
last  example,  or  when  the  exponent  contains  a decimal,  as 
in  several  instances  in  the  examples  given  on  pages  45-49. 
Such  examples  can  be  solved  without  the  use  of  logarithms, 
but  the  process  is  very  long  and  somewhat  involved,  and  the 
arithmetical  work  required  is  enormous.  To  solve  the  exam- 
ple last  given  without  using  the  logarithmic  table  and  obtain 
the  value  of  x correct  to  five  figures  would  require,  perhaps, 
100  times  as  many  figures  as  were  used  in  the  solution  given, 
and  the  resulting  liability  to  error  would  be  correspondingly 
increased;  indeed,  to  confine  the  work  to  this  number  of 
figures  would  also  require  a good  knowledge  of  short-cut 
methods  in  multiplication  and  division,  and  judgment  and 
skill  on  the  part  of  the  calculator  that  can  only  be  acquired 
by  practice  and  experience. 

Formulas  containing  quantities  affected  with  decimal 
exponents  are  generally  of  an  empirical  nature;  that  is,  the 
constants  or  exponents  or  both  are  given  such  values  as  will 
make  the  results  obtained  by  the  formulas  agree  with  those 
obtained  by  experiment.  Such  formulas  occur  frequently  in 
works  treating  on  thermodynamics,  strength  of  materials, 
machine  design,  etc. 


50 


USEFUL  TABLES. 


COMMON  LOGARITHMS. 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

400 

00  000 

043 

087 

130 

173 

217 

260 

303 

346 

“389 

101 

432 

475 

518 

561 

604 

647 

“689 

“732 

~m> 

“817 

44 

43 

42 

102 

860 

903 

945 

988 

*030 

*072 

*115 

*157 

*199 

*242 

1 

4.4 

4.3 

4.2 

103 

01  284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

2 

8.8 

8.6 

8.4 

104 

703 

745 

787 

828 

870 

912 

953 

995 

*036 

*078 

3 

13.2 

12.9 

12.6 

105 

02  119 

160 

202 

243 

284 

325 

366 

407 

4 19 

490 

4 

17.6 

17.2 

16.8 

106 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

5 

22.0 

21.5 

21.0 

107 

938 

979 

*019 

*060 

*100 

*141 

*181 

*222 

*262 

*302 

6 

26.4 

25.8 

25.2 

108 

03  342 

383 

423 

463 

503 

543 

583 

623 

663 

703 

7 

30.8 

30.1 

29.4 

109 

743 

782 

822 

862 

902 

941 

981 

*021 

*060 

*100 

81 

35.2 

34.4 

33.6 

4io; 

04  139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

9 

39.6 

38.7 

37.8 

111 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

41 

40 

39 

112 

922 

961 

999 

*038 

*077 

*115 

*154 

*192 

*231 

*269 

1 

4.1 

4.0 

3.9 

113 

05  308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

2 

8.2 

8.0 

7.8 

114 

690 

729 

767 

805 

843 

881 

918 

956 

994 

*032 

3 

12.3 

12.0 

11.7 

115 

06  070 

108 

145 

183 

221 

258 

296 

333 

371 

408 

4 

16.4 

16.0 

15.6 

116 

446 

483 

521 

558 

595 

633 

670 

707 

744 

781 

5 

20.5 

20.0 

19.5 

117 

819 

856 

893 

930 

967 

*004 

•nil 

*078 

*115 

*151 

6 

24.6 

24.0 

23.4 

118 

07  188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

7 

28.7 

28.0 

27.3 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

8 

32.8 

32.0 

31.2 

420 

918 

954 

990 

*027 

*063 

*099 

*135 

*m 

*207 

*243 

9 

36.9 

36.0 

35.1 

121 

08  279 

314 

350 

386 

422 

458 

493 

529 

565 

600 

38 

37 

36 

122 

636 

672 

707 

743 

778 

814 

849 

884 

920 

955 

1 

3.8 

3.7 

3.6 

123 

991 

*026 

*061 

*096 

*132 

*167 

*202 

*237 

*272 

*307 

2 

7.6 

7.4 

7.2 

124 

09  342 

377 

412 

447 

482 

517 

552 

587 

621 

65(3 

3 

11.4 

11.1 

10.8 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968 

*003 

4 

15.2 

14.8 

14.4 

126 

10  037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

5 

19.0 

18.5 

18.0 

127 

380 

415 

449 

483 

517 

551 

585 

619 

653 

687 

6 

22.8 

22.2 

21.6 

128 

721 

755 

789 

823 

857 

890 

924 

958 

992 

*025 

7 

26.6 

25.9 

25.2 

129 

11  059 

093 

126 

160 

l:»:; 

227 

261 

294 

327 

361 

8 

9 

30.4 

34.2 

29.6 

33.3 

28.8 

32.4 

430 

394 

428 

461. 

494 

528 

561 

594 

628 

661 

694 

131 

727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 

35 

34 

33 

132 

12  057 

090 

123 

156 

189 

222 

254 

287 

320 

352 

1 

3.5 

3.4 

3.3 

133 

385 

418 

450 

483 

516 

548 

581 

613 

646 

67S 

2 

7.0 

6.8 

6.6 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

*001 

3 

10.5 

10.2 

9.9 

135 

13  033 

066 

098 

130 

162 

194 

226 

258 

290 

322 

4 

14.0 

13.6 

13.2 

136 

354 

386 

418 

450 

481 

513 

545 

577 

(309 

640 

5 

17.5 

17.0 

16.5 

137 

672 

704 

735 

767 

799 

830 

862 

893 

925 

956 

6 

21.0 

20.4 

19.8 

138 

988 

*019 

*051 

*082 

*114 

*145 

*176 

*208 

*239 

*270 

7 

24.5 

23.8 

23.1 

139 

14  301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

8 

28.0 

31.5 

27.2 

30.6 

26.4 

29.7 

440 

613 

644 

675 

706 

737 

768 

799 

829 

860 

891 

9 

141 

922 

953 

983 

*014 

*045 

*076 

*106 

*137 

*168 

*198 

32 

31 

30 

142 

15  229 

259 

200 

320 

351 

381 

412 

442 

473 

503 

1 

3.2 

3.1 

3.0 

143 

534 

564 

594 

625 

655 

6s  5 

715 

746 

776 

806 

2 

6.4 

6.2 

6.0 

144 

836 

866 

897 

927 

957 

987 

*017 

*047 

*077 

*107 

3 

9.6 

9.3 

9.0 

145 

16  137 

167 

197 

227 

256 

286 

316 

346 

376 

406 

4 

12.8 

12.4 

12.0 

146 

435 

465 

495 

524 

554 

584 

613 

643 

673 

702 

5 

16.0 

15.5 

15.0 

147 

732 

761 

791 

820 

850 

879 

909 

938 

967 

997 

6 

19.2 

18.6 

18.0 

148 

17  026 

056 

085 

114 

143 

173 

202 

231 

260 

289 

7 

22.4 

21.7 

21.0 

149 

319 

348 

377 

406 

435 

464 

493 

522 

551 

580 

8 

25.6 

24.8 

24.0 

28.8 

27.9 

27.0 

450 

609 

638 

667 

696 

725 

754 

782 

811 

840 

”869 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

LOGARITHMS. 


51 


Table — ( Continued). 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

150 

17  609 

638 

667 

696 

725 

754 

“782 

811 

840 

869 

151 

898 

'926 

955 

“984 

#013 

*041 

*070 

*099 

*127 

*156 

29 

28 

152 

18  184 

213 

241 

270 

298 

327 

355 

384 

112 

441 

l 

2.9  1 

2.8 

153 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

2 

5.8 

5.6 

154 

752 

780 

808 

837 

865 

893 

921 

949 

977 

*005 

3 

8.7 

8.4 

155 

19  033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

4 

11.6  1 

11.2 

156 

312 

340 

368 

396 

424 

45L 

479 

507 

535 

562 

5 

14.5  ; 

14.0 

157 

590 

618 

645 

673 

700 

728 

756 

783 

811 

838 

6 

17 

'.4 

16.8 

158 

866 

893 

921 

948 

976 

*003 

*030 

*058 

*085 

*112 

7 

20.3 

19.6 

159 

20  140 

167 

194 

222 

249 

276 

303 

330 

358 

385 

8 

23.2 

22.4 

160 

412 

439 

466 

493 

520 

548 

“575 

“602 

629 

“656 

9 

26.1 

25.2 

161 

683 

710 

737 

763 

790 

817 

844 

871 

898 

925 

27 

26 

162 

952 

978 

*005 

*032 

*059 

*085 

*112 

*139 

*165 

*192 

1 

2.7  1 

1 2.6 

163 

21  219 

245 

272 

299 

325 

352 

378 

405 

431 

458 

2 

5.4 

1 5.2 

164 

484 

511 

537 

564 

590 

617 

643 

669 

696 

722 

3 

8.1 

| 7.8 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

4 

10.8 

| 10.4 

166 

22  Oil 

037 

063 

089 

115 

141 

167 

194 

220 

246 

5 

13.5 

i 13.0 

167 

272 

298 

324 

350 

376 

401 

427 

453 

479 

505 

6 

16.2 

i 15.6 

168 

531 

557 

583 

608 

634 

660 

686 

712 

737 

763 

7 

18.9 

1 18.2 

169 

789 

814 

840 

866 

891 

917 

943 

968 

994 

*019 

8 

21.6 

1 20.8 

i 23.4 

170 

23  045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

171 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

25 

172 

553 

578 

603 

629 

654 

679 

704 

729 

754 

779 

1 

2.5 

173 

805 

830 

855 

880 

905 

930 

955 

980 

■ini:, 

*030 

2 

5.0 

174 

24  055 

080 

105 

130 

155 

180 

204 

229 

254 

279 

4 

7.5 

175 

304 

329 

353 

378 

403 

428 

452 

477 

502 

527 

4 

10.0 

176 

551 

576 

601 

625 

()50 

674 

699 

724 

748 

773 

5 

12.5 

177 

797 

822 

846 

871 

895 

920 

944 

969 

993 

*018 

6 

15.0 

178 

25  042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

7 

17.5 

179 

285 

310 

334 

358 

382 

406 

431 

455 

479 

503 

8 

20.0 

180 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

£4  *9 

181 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24 

23 

182 

26  007 

031 

055 

079 

102 

126 

150 

174 

198 

221 

1 

2.4 

1 2.3 

183 

245 

269 

293 

316 

340 

364 

387 

411 

435 

458 

2 

4.8 

4.6 

184 

482 

505 

529 

553 

576 

600 

623 

647 

670 

694 

3 

7.2 

6.9 

185 

717 

741 

764 

788 

811 

834 

858 

881 

905 

928 

4 

c 

>.6 

9.2 

186 

951 

975 

998 

*021 

*045 

*068 

*091 

*114 

*138 

*161 

5 

12.0 

| 11.5 

187 

27  184 

207 

231 

254 

277 

300 

323 

346 

370 

393 

6 

14.4 

! 13.8 

188 

416 

439 

462 

485 

508 

531 

554 

577 

600 

623 

7 

16.8 

! 16.1 

189 

646 

669 

692 

715 

738 

761 

784 

807 

830 

852 

8 

19.2 

18.4 

190 

875 

”898 

"921 

“944 

“967 

989 

*012 

*035 

*058 

*081 

9 

21.6 

[ 20.7 

191 

28  103 

~126 

~149 

“m 

194 

“217 

“240 

“262 

285 

“307 

22 

21 

192 

330 

353 

375 

398 

421 

443 

466 

488 

511 

533 

1 

2.2 

2.1 

193 

556 

578 

601 

623 

646 

668 

691 

713 

735 

758 

2 

4.4 

4.2 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

3 

< 

5.6 

6.3 

195 

29  003 

026 

048 

070 

092 

115 

137 

159 

181 

203 

4 

8.8 

8.4 

196 

226 

248 

270 

292 

31 1 

336 

358 

380 

403 

425 

5 

11.0 

10.5 

197 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

6 

13.2 

12.6 

198 

667 

68> 

710 

732 

754 

776 

798 

820 

842 

863 

7 

15.4 

14.7 

199 

885 

907 

929 

951 

973 

994 

*016 

*038 

*060 

*081 

8 

r 

i.6 

16.8 

200 

30  103 

“l25 

“l46 

“l68 

“l90 

“2U 

“233 

“255 

“276 

“298 

9 

13.8 

18.9 

N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

52 


USEFUL  TABLES. 


Table — ( Continued). 


N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

r, 

. r 

200 

30  103 

125 

146 

168 

190 

211 

233 

255 

276 

298 

201 

320 

341 

363 

384 

406 

428 

449 

471 

492 

"514 

22 

21 

202 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

1 

2.2 

2ul 

203 

750 

771 

792 

814 

835 

856 

878 

899 

920 

942 

2 

4.4 

4.2 

204 

963 

984 

*006 

*027 

*048 

*069 

*091 

*112 

*133 

*154 

3 

1 

5.6 

6.3 

205 

31  175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

4 

8.8 

8.4 

206 

387 

408 

429 

450 

471 

492 

513 

534 

555 

576 

5 

11.0 

10.5 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

6 

13.2 

12.6 

208 

806 

827 

848 

869 

890 

911 

931 

952 

973 

994 

7 

15.4 

14:7 

209 

32  015 

035 

056 

077 

098 

118 

139 

160 

181 

201 

8 

17.6 

16.8 

210 

222 

243 

263 

284 

305 

~325 

"346 

~366 

"387 

"408 

9 

19.8 

18.9 

211 

428 

449 

469 

490 

510 

531 

552 

572 

593 

613 

20 

212 

634 

654 

675 

695 

715 

736 

756 

777 

797 

818 

1 

2.0 

213 

838 

858 

879 

899 

9l9 

940 

960 

980 

*001 

*021 

2 

4.0 

214 

33  041 

062 

082 

102 

122 

143 

163 

183 

203 

224 

3 

6.0 

215 

244 

264 

284 

304 

325 

345 

365 

385 

405 

425 

4 

8.0 

216 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

5 

10.0 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

6 

12.0 

218 

846 

866 

885 

905 

925 

945 

965 

985 

*005 

*025 

7 

14.0 

219 

34  044 

064 

084 

104 

124 

143 

163 

183 

203 

223 

8 

16.0 

220 

242 

262 

282 

301 

321 

341 

361 

380 

400 

420 

18.0 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

19 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

1 

1.9 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

*005 

2 

3.8 

224 

35  025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

3 

5.7 

225 

218 

238 

257 

276 

295 

315 

334 

353 

372 

392 

4 

7.6 

226 

411 

430 

449 

468 

488 

507 

526 

545 

564 

583 

5 

9.5 

227 

603 

622 

641 

660“ 

679 

698 

717 

736 

755 

774 

6 

11.4 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

7 

13.3 

229 

984 

*003 

*021 

*040 

*059 

*078 

*097 

*116 

*135 

*154 

8 

15.2 

230 

36  173 

192 

211 

229 

248 

267 

286 

305 

324 

342 

9 

17.1 

231 

361 

380 

399 

418 

436 

455 

474 

493 

511 

530 

18 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

1 

1.8 

233 

736 

754 

773 

791 

810 

829 

847 

806 

884 

903 

2 

3.6 

234 

922 

940 

959 

977 

996 

*014 

*033 

•051 

*070 

*088 

3 

5.4 

235 

37  107 

125 

144 

162 

181 

199 

218 

236 

254 

273 

4 

7.2 

236 

291 

310 

328 

346 

365 

383 

401 

420 

43s 

457 

5 

9.0 

237 

475 

493 

511 

530 

548 

566 

585 

603 

621 

639 

6 

10.8 

238 

658 

676 

694 

712 

731 

749 

767 

785 

803 

822 

7 

12.6 

239 

840 

858 

876 

894 

912 

931 

949 

967 

985 

*003 

8 

14.4 

240 

38  021 

039 

057 

075 

093 

112 

130 

148 

166 

"184 

9 

16.2 

241 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

17 

242 

382 

399 

417 

435 

453 

471 

489 

507 

525 

543 

1 

1.7 

243 

561 

578 

596 

614 

632 

650 

668 

686 

703 

721 

2 

3.4 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

3 

5.1 

245 

917 

934 

952 

970 

987 

*005 

*023 

'041 

*058 

*076 

4 

6.8 

246 

39  094 

111 

129 

146 

164 

182 

199 

217 

235 

252 

5 

8.5 

247 

270 

287 

305 

322 

340 

358 

375 

393 

410 

428 

6 

10.2 

248 

445 

463 

480 

498 

515 

533 

550 

568 

585 

602 

7 

11.9 

249 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

8 

9 

13.6 

250 

794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P. 

P. 

LOGARITHMS. 


53 


Table— ( Continued). 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 

250 

39  794 

811 

829 

846 

863 

881 

898 

915 

933 

950 

251 

967 

985 

*002 

*019 

*037 

*054 

*071 

*088 

*106 

*123 

18 

252 

40  140 

157 

175 

192 

209 

226 

243 

261 

27* 

295 

1 

1.8 

253 

312 

329 

346 

364 

381 

398 

415 

432 

449 

466 

2 

3.6 

254 

483 

500 

518 

535 

552 

569 

586 

603 

620 

637 

3 

5.4 

255 

654 

671 

688 

705 

722 

739 

756 

773 

790 

807 

4 

7.2 

256 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

i 5 

9.0 

257 

993 

*010 

*027 

*044 

*061 

*078 

*095 

*111 

*12S 

*145 

6 

10.8 

258 

41  16^ 

179 

196 

212 

229 

246 

263 

280 

296 

313 

7 

12.6 

259 

33(f 

347 

363 

380 

397 

414 

430 

447 

464 

481 

8 

9 

14.4 

260 

497 

514 

531 

547 

564 

581 

597 

614 

631 

647 

16.2 

261 

664 

681 

697 

714 

731 

747 

764 

780 

797 

814 

17 

262 

830 

847 

863 

880 

896 

913 

929 

946 

963 

979 

1 

1.7 

263 

996 

*012 

*029 

*045 

*062 

*078 

*095 

*111 

*127 

•144 

2 

3.4 

264 

42  160 

177 

193 

210 

226 

243 

259 

275 

292 

308 

1 3 

5.1 

265 

325 

341 

357 

374 

39C 

406 

423 

439 

455 

472 

4 

6.8 

266 

488 

504 

521 

537 

553 

570 

586 

602 

619 

635 

5 

8.5 

267 

651 

667 

684 

700 

716 

732 

749 

765 

781 

797 

6 

10.2 

268 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

7 

11.9 

269 

975 

991 

*008 

*024 

*040 

*056 

*072 

*088 

*104 

*120 

8 

13.6 

270 

43  136 

152 

169 

185 

201 

217 

233 

249 

265 

281 

9 

15.3 

271 

297 

313 

329 

345 

361 

377 

393 

409 

425 

441 

16 

272 

457 

473 

489 

505 

521 

537 

553 

569 

584 

600 

1 

1.6 

273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

2 

3.2 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

3 

4.8 

275 

933 

949 

965 

9M 

*012 

*028 

*044 

*059 

*075 

4 

6.4 

276 

44  091 

107 

122 

138 

154 

170 

185 

201 

217 

232 

5 

8.0 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

6 

9.6 

278 

404 

420 

436 

451 

467 

483 

498 

514 

529 

545 

7 

11.2 

279 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

8 

12.8 

280 

716 

731 

747 

762 

778 

793 

809 

824 

840 

855 

9 

14.4 

281 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*010 

15 

282 

45  025 

040 

056 

071 

086 

102 

117 

133 

148 

163 

1 

1.5 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

317 

2 

3.0 

284 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

3 

4.5 

285 

484 

500 

515 

530 

545 

561 

576 

591 

606 

621 

4 

6.0 

286 

637 

652 

667 

0,-2 

697 

712 

728 

743 

758 

773 

5 

7.5 

287 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

6 

9.0 

288 

939 

954 

969 

984 

*000 

*015 

*030 

*045 

*060 1 

*075 

7 

10.5 

289 

46  090 

105 

120 

135 

15U 

165 

180 

195 

210 

225 

8 

9 

12.0 

13.5 

290 

240 

255 

270 

285 

300 

315 

330 

345 

359 

374 

291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

14 

292 

538 

553 

568 

583 

598 

613 

627 

642 

657 

672 

1 

1.4 

293 

687 

702 

716 

731 

746 

761 

776 

790 

805 

820 

2 

2.8 

294 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

3 

4.2 

295 

982 

997 

*012 

*026 

*041 

*056 

*070 

*085 

*100 

*114 

4 

5.6 

296 

47  129 

144 

159 

173 

188 

202 

217 

232 

246 

261 

5 

7.0 

297 

276 

290 

305 

319 

334 

349 

363 

378 

392 

407 

6 

8.4 

298 

422 

436 

451 

465 

480 

494 

509 

5241 

538 

553 

7 

9.8 

299 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

8 

11.2 

300 

712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

9 

12.6 

N. 

L.  0 

1* 

2 

3 

4 

5 1 

6 

7 

8 

9 

P. 

P. 

54 


USEFUL  TABLES. 


Table — ( Continued). 


LOGARITHMS. 


55 


Table— ( Continued). 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

350 

851 

352 

353 

354 

355 

356 

357 

358 

359 

360 

361 

362 

363 

364 

365 

366 

367 

368 

369 

370 

371 

372 

373 

374 

375 

376 

377 

378 

379 

380 

381 

382 

383 

384 

385 

386 

387 

388 

389 

390 

391 

392 

393 

394 

395 

396 

397 

398 

399 

400 

N. 

54  407 

419 

“543 

667 

790 

913 

035 

157 

279 

400 

522 

432 

“555 

679 

802 

925 

047 

169 

291 

413 

534 

444 

"568 

691 

814 

937 

060 

182 

303 

42a 

546 

456 

5si) 

704 

827 

949 

072 

194 

315 

437 

558 

469 

“593 

716 

839 

962 

(tsi 

2H6 

328 

449 

570 

481 

“605 

728 

851 

974 

096 

218 

340 

461 

582 

494 

“617 

741 

864 

986 

108 

230 

352 

473 

594 

“506 

“630 

753 

876 

998 

121 

242 

364 

485 

606 

518 

“642 

765 

888 

*011 

133 

255 

376 

497 

618 

13 

1 1.3 

2 2.6 

3 3.9 

4 5.2 

5 6.5 

6 7.8 

7 9.1 

8 10.4 

9 11.7 

12 

1 1.2 

2 2.4 

3 3.6 

4 4.8 

5 6.0 

6 7.2 

7 8.4 

8 9.6 

9 10.8 

II 

1 1.1 

2 2.2 

3 3.3 

4 4.4 

5 5.5 

6 6.6 

7 7.7 

8 8.8 

9 9.9 

10 

1 1.0 

2 2.0 

3 3.0 

4 4.0 

5 5.0 

6 6.0 

7 7.0 

8 8 0 

9 9.0 

531 
654 
777 
900 
55  023 
145 
267 
388 
509 

630 

642 

654 

666 

678 

691 

703 

715 

727 

739 

“859 

979 

*098 

217 

336 

455 

573 

691 

808 

751 
871 
991 
56  110 
229 
348 
467 
585 
703 

763 

883 

*003 

122 

241 

360 

478 

597 

714 

775 

895 

*015 

134 

253 

372 

490 

608 

726 

787 

907 

*027 

146 

265 

384 

502 

620 

738 

799 

919 

*038 

158 

277 

396 

514 

632 

750 

811 

931 

*050 

170 

289 

407 

526 

644 

761 

823 

943 

*062 

182 

301 

419 

538 

656 

773 

835 

955 

*074 

194 

312 

431 

549 

667 

785 

847 

967 

*086 

205 

324 

443 

561 

679 

797 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

937 
57  054 
171 
287 
403 
519 
634 
749 
864 

949 

066 

183 

299 

415 

530 

646 

761 

875 

961 

078 

194 

310 

426 

542 

657 

772 

887 

972 

089 

206 

322 

438 

553 

669 

784 

898 

984 

101 

217 

334 

449 

565 

680 

795 

910 

996 

113 

229 

345 

461 

576 

692 

807 

921 

*008 

124 

241 

357 

473 

588 

703 

818 

933 

*019 

136 

252 

368 

484 

600 

715 

830 

944 

*031 

148 

264 

3S0 

496 

611 

726 

841 

955 

*043 

159 

276 

392 

507 

623 

738 

852 

967 

978 
58  092 
206 
320 
433 
546 
659 
771 
883 
995 

990 

104 

218 

331 

444 

557 

670 

782 

894 

*006 

*001 

115 

229 

343 

456 

569 

681 

794 

906 

*017 

*013 

127 

240 

354 

467 

580 

692 

805 

917 

*028 

*024 

138 

252 

365 

478 

591 

704 

816 

928 

*040 

*035 

149 

263 

377 

490 

602 

715 

827 

939 

*051 

*047 

161 

274 

388 

501 

614 

726 

838 

950 

*062 

*058 

172 

286 

399 

512 

625 

737 

850 

961 

*073 

*070 

184 

297 

410 

524 

636 

749 

861 

973 

*084 

*081 

195 

309 

422 

535 

647 

760 

872 

984 

*095 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

218 
329 
439 
550 
660 
770 
879 
988 
60  097 

229 

340 

450 

561 

671 

780 

890 

999 

108 

240 

351 

461 

572 

682 

791 

901 

*010 

119 

251 

362 

472 

583 

693 

802 

912 

*021 

130 

262 

373 

483 

594 

704 

813 

923 

*032 

141 

273 

384 

494 

605 

715 

824 

934 

*043 

152 

284 

395 

506 

616 

726 

835 

945 

*054 

163 

295 

406 

517 

627 

737 

846 

956 

*065 

173 

306 

417 

528 

638 

748 

857 

966 

*076 

184 

318 

428 

539 

649 

759 

868 

977 

*086 

195 

206 

L.  0 

217 

228 

239 

249 

260 

271 

282 

293 

304 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

56 


USEFUL  TABLES. 


Table — ( Continued). 


N. 

L.O 

1 

2 

3 

4 

*5 

6 

7 

8 

9 

P.  P. 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

304 

401 

314 

325 

336 

347 

369 

“379 

"390 

^401 

“412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

509 

520 

403 

531 

541 

552 

563 

574 

584 

595 

606 

617 

627 

404 

638 

649 

660 

670 

681 

692 

703 

713 

724 

735 

405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

II 

407 

959 

970 

981 

991 

*002 

*013 

*023 

*034 

*045 

*055 

1 

1.1 

408 

61  066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

2 

2.2 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

3 

3.3 

410 

278 

289 

300 

310 

321 

331 

342 

352 

363 

374 

4 

5 

4.4 

5.5 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

6 

6.6 

412 

490 

500 

511 

521 

532 

542 

553 

563 

574 

584 

J7 

7.7 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

8 

8.8 

414 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

9 

9.9 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

416 

909 

920 

930 

941 

951 

962 

972 

982 

993 

*003 

417 

62  014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

418 

118 

128 

138 

149 

159 

170 

180 

190 

201 

211 

419 

221 

232 

242 

252 

263 

273 

284 

294 

304 

315 

420 

325 

335 

346 

356 

366 

377 

"387 

“397 

~408 

~418 

10 

421 

428 

439 

449 

459 

469 

480 

490 

500 

511 

521 

422 

531 

542 

552 

562 

572 

583 

503 

603 

613 

624 

1 

1.0 

423 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

2 

2.0 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

3 

3.0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

931 

4 

4.0 

426 

941 

951 

961 

972 

982 

902 

*002 

*012 

*022 

*033 

5 

5.0 

427 

63  043 

053 

063 

073 

083 

094 

104 

114 

124 

134 

6 

6.0 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

7 

7.0 

429 

246 

256 

266 

276 

286 

296 

306 

317 

327 

337 

8 

g 

8.0 

9.0  j 

430 

347 

357 

367 

377 

387 

397 

407 

417 

428 

~438 

431 

448 

458 

468 

478 

488 

498 

508 

518 

528 

538 

432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

434 

749 

759 

769 

779 

789 

799 

809 

819 

829 

839 

435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

436 

949 

959 

969 

979 

988 

998 

*008 

*018 

*028 

*038 

9 

437 

64  048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

1 

0.9 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

2 

1.8 

439 

246 

256 

266 

276 

286 

296 

306 

316 

326 

335 

3 

2.7 

440 

345 

"355 

“365 

~375 

385 

"395 

~404 

414 

424 

~434 

4 

5 

3.6 

441 

444 

454 

464 

473 

483 

493 

503 

513 

523 

532 

6 

5.4 

442 

542 

552 

562 

572 

582 

591 

601 

611 

621 

631 

7 

6.3 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

g 

7.2 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

g 

8.1 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 

446 

933 

943 

953 

963 

972 

982 

992 

*002 

*011 

*021 

447 

65  031 

040 

050 

060 

070 

079 

089 

099 

108 

118 

448 

128 

137 

147 

157 

167 

176 

186 

196 

205 

215 

449 

225 

234 

244 

254 

263 

273 

283 

292 

302 

312 

450 

321 

331 

341 

350 

360 

369 

379 

389 

398 

408 

N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

LOGARITHMS. 


67 


Table— ( Continued). 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

450 

65  321 

331 

341 

350 

360 

”369 

”379 

”389 

”398 

408 

451 

418 

427 

437 

447 

456 

"466 

175 

”485 

"495 

”504 

452 

514 

523 

533 

543 

552 

562 

571 

581 

591 

600 

453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

454 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 

455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

457 

992 

*001 

*011 

*020 

*030 

*039 

*049 

*058 

*068 

*077 

458 

66  087 

096 

106 

115 

124 

134 

143 

153 

162 

172 

459 

181 

191 

200 

210 

219 

229 

238 

247 

257 

266 

460 

276 

285 

295 

304 

314 

323 

332 

342 

351 

”361 

461 

370 

380 

389 

398 

408 

417 

427 

436 

445 

455 

462 

464 

474 

483 

492 

502 

511 

521 

530 

539 

549 

463 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

467 

932 

941 

950 

960 

969 

978 

987 

997 

*006 

*015 

468 

67  025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

471 

302 

~3U 

321 

330 

339 

“348 

~357 

”367 

"376 

”385 

472 

394 

403 

413 

422 

431 

440 

449 

459 

468 

477 

473 

486 

495 

504 

514 

523 

532 

541 

550 

560 

569 

474 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

475 

669 

679 

688 

697 

706 

7]  5 

724 

733 

742 

752 

476 

761 

770 

779 

788 

797 

800 

815 

825 

834 

843 

477 

852 

861 

870 

879 

888 

897 

906 

916 

925 

934 

478 

943 

952 

961 

970 

979 

988 

997 

*006 

*015 

*024 

479 

68  034 

043 

052 

061 

070 

079 

088 

097 

106 

115 

480 

124 

133 

142 

151 

160 

169 

178 

187 

196 

205 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

305 

314 

323 

332 

341 

350 

359 

368 

377 

386 

483 

395 

404 

413 

422 

431 

440 

449 

458 

467 

476 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

488 

842 

851 

860 

869 

878 

886 

895 

904 

913 

922 

489 

931 

940 

949 

958 

966 

975 

984 

993 

*002 

*011 

490 

69  020 

028 

037 

046 

055 

064 

073 

082 

090 

099 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

188 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

494 

373 

381 

390 

399 

408 

417 

425 

434 

443 

452 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

497 

636 

644 

653 

662 

671 

679 

'688 

697 

705 

714 

498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 

500 

897 

”906 

914 

”923 

”932 

”940 

”949 

”958 

”966 

“975 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 


10 

1 1.0 
2 2.0 

3 3.0 

4 4.0 

5 5.0 

6 6.0 

7 7.0 

8 8.0 

9 9.0 


9 

1 0.9 

2 1.8 

3 2.7 

4 3.6 

5 4.5 

6 5.4 

7 6.3 

8 7.2 

9 8.1 


8 

1 0.8 

2 1.6 

3 2.4 

4 3.2 

5 4.0 

6 4.8 

7 5.6 

8 6.4 

9 7.2 


P.  P. 


58 


USEFUL  TABLES. 


Table— ( Continued ). 


N. 

L.  0 

1 

2- 

3 

4 

5 1 

6 1 

7 

8 1 

9 

P.  P. 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

501 

984 

992 

*001 ,*010 

*018 

*027 

*036  *044 

*053  *062 

502 

70  070 

079 

088 

096 

105 

114 

122 

131 

140 

148 

503 

157 

165 

174 

183 

191 

200 

209 

217 

226 

234 

504 

243 

252 

260 

269 

278 

286 

295 

303 

312 

321 

505 

329 

338 

346 

355 

364 

372 

381 

389 

398 

406 

506 

415 

424 

432 

441 

449 

458 

467 

475 

484 

492 

9 

507 

501 

509 

518 

526 

535 

544 

552 

561 

569 

578 

1 

0.9 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

2 

1.8 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

3 

2.7 

510 

757 

766 

'774 

783 

791 

800 

“808 

”817 

“825 

”834 

4 

5 

3.6 

4.5 

511 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

g 

5.4 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995 

*003 

6.3 

513 

71  012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

g 

7.2 

514 

096 

105 

113 

122 

130 

139 

147 

155 

164 

172 

9 

g|l 

515 

181 

189 

198 

206 

214 

223 

231 

240 

248 

257 

516 

265 

273 

282 

290 

299 

307 

315 

324 

332 

341 

517 

349 

357 

366 

374 

383 

391 

399 

408 

416 

425 

518 

433 

441 

450 

458 

466 

475 

483 

492 

500 

508 

519 

517 

525 

533 

542 

550 

559 

567 

575 

584 

592 

520 

600 

609 

617 

625 

634 

642 

650 

”659 

667 

675 

521 

684 

692 

700 

709 

717 

725 

734 

742 

7 50 

759 

8 

522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

1 

0.8 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

2 

1.6 

524 

933 

941 

950 

958 

966 

975 

983 

991 

999 

*008 

3 

2.4 

525 

72  016 

024 

032 

041 

049 

057 

066 

074 

082 

090 

4 

3.2 

526 

099 

107 

115 

123 

132 

140 

148 

156 

165 

173 

5 

4.0 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

6 

4.8 

528 

263 

272 

280 

288 

296 

304 

313 

321 

329 

337 

7 

5.6 

529 

346 

354 

362 

370 

378 

387 

395 

403 

411 

419 

8 

6.4 

530 

428 

~436 

"444 

“452 

“460 

“469 

“477 

485 

“493 

”501 

9 

7.2 

531 

509 

"518 

“526 

~534 

"542 

“550 

“558 

“567 

“575 

583 

532 

591 

599 

607 

616 

624 

632 

640 

648 

656 

665 

533 

673 

681 

689 

697 

705 

713 

722 

730 

738 

746 

534 

754 

762 

770 

779 

787 

795 

803 

! 811 

819:  827 

535 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 

536 

916 

925 

933 

941 

949 

957 

965 

! 973 

981 

989 

7 

• 537 

997 

*006 

*014 

*022 

*030 

*038 

*046  *054 

*062 

*070 

1 

0.7 

538 

73  078 

086 

*094 

102 

111 

119 

127 

135 

143 

151 

2 

1.4 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

3 

2.1 

540 

239 

247 

255 

263 

272 

280 

288 

[”296 

“304 

”312 

4 

2.8 

541 

320 

”328 

336 

”344 

“352 

“360 

“368 

“376 

”384 

”392 

g 

4.2 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

4.9 

543 

480 

488 

496 

| 504 

512 

520 

528 

536 

544 

552 

8 

5.6 

544 

560 

568 

576 

! 584 

592 

600 

608 

616 

624 

632 

9 

6.3 

545 

640 

648 

656 ! 664 

672 

679 

687 

695 

703 

711 

546 

719 

727 

735 

1 743 

751 

759 

767 

775 

783 

791 

547 

799 

807 

815 

823 

830 

838 

846 

8o4 

862 

870 

548 

878 

886 

894 

! 902 

910 

918 

926 

933 

941 

949 

549 

957 

965 

973 

981 

989 

997 

*005, *013 

*020  *028 

550 

?4  036 

t044 

“052 

060  068 

“076 

| 084 

| 092 

099 

107 

N. 

L.O 

1 

2 

i 3 

i 4 

5 

1 6 

7 

8 

9 

l 

P.  P. 

N. 

50 

551 

552 

553 

554 

555 

556 

557 

558 

559 

60 

561 

562 

563 

564 

565 

566 

567 

568 

569 

70 

571 

572 

573 

574 

575 

576 

577 

578 

579 

80 

581 

582 

583 

584 

585 

586 

587 

588 

589 

90 

591 

592 

593 

594 

595 

596 

597 

598 

599 

00 

N. 


LOGARITHMS. 


59 


Table— ( Continued). 


2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

052 

060 

068 

076 

084 

092 

099 

107 

131 

139 

147 

155 

162 

170 

178 

186 

210 

218 

225 

233 

241 

249 

257 

265 

288 

296 

304 

312 

320 

327 

335 

343 

367 

374 

382 

390 

398 

406 

414 

421 

445 

453 

461 

468 

476 

484 

492 

500 

523 

531 

539 

547 

554 

562 

570 

578 

601 

609 

617 

624 

632 

640 

648 

656 

679 

687 

695 

702 

710 

718 

726 

733 

757 

764 

772 

780 

788 

796 

803 

811 

834 

842 

850 

858 

865 

873 

881 

889 

912 

920 

927 

935 

943 

950 

958 

966 

8 

989 

997 

*005 

*012 

*020 

*028 

*035 

*043 

1 

0.3 

066 

074 

082 

089 

097 

105 

113 

120 

2 

1.6 

143 

151 

159 

166 

174 

182 

189 

197 

3 

2.4 

220 

228 

236 

243 

251 

259 

266 

274 

4 

3.2 

297 

305 

312 

320 

32* 

335 

343 

351 

5 

4.0 

374 

381 

289 

397 

404 

412 

420 

427 

6 

4.8 

450 

458 

465 

473 

481 

488 

496 

504 

7 

5.6 

526 

534 

542 

549 

557 

565 

572 

580 

8 

9 

6.4 

7.2 

603 

610 

618 

626 

633 

641 

648 

656 

679 

686 

694 

702 

709 

717 

724 

732 

755 

762 

770 

778 

785 

793 

800 

808 

831 

838 

846 

853 

861 

868 

876 

884 

906 

914 

921 

929 

937 

944 

952 

959 

982 

989 

997 

*005 

*012 

*020 

"627 

*035 

057 

065 

072 

080 

087 

095 

103 

110 

133 

140 

148 

155 

163 

170 

178 

185 

208 

215 

223 

230 

238 

245 

253 

260 

283 

290 

298 

305 

313 

320 

328 

335 

358 

365 

373 

380 

388 

395 

403 

410 

433 

440 

448 

455 

462 

470 

477 

485 

7 

507 

515 

522 

530 

537 

545 

552 

559 

1 

0.7 

582 

589 

597 

604 

612 

619 

626 

634 

2 

1.4 

656 

664 

671 

678 

686 

693 

701 

708 

3 

2.1 

730 

738 

745 

753 

760 

768 

775 

782 

4 

2.8 

805 

812 

819 

827 

834 

842 

849 

856 

5 

3.5 

879 

886 

893 

901 

908 

916 

923 

930 

6 

4.2 

953 

960 

967 

975 

982 

989 

997 

*004 

7 

4.9 

026 

034 

041 

048 

056 

063 

070 

078 

8 

5.6 

100 

107 

115 

122 

129 

137 

144 

151 

9 

6.3 

173 

181 

188 

195 

203 

210 

217 

225 

247 

254 

262 

269 

276 

283 

291 

298 

320 

327 

335 

342 

349 

357 

364 

371 

393 

401 

408 

415 

422 

430 

437 

444 

466 

474 

481 

488 

495 

503 

510 

517 

539 

546 

554 

561 

568 

576 

5831 

590 

612 

619 

627 

634 

641 

648 

656| 

663 

685 

692 

699 

706 

714 

721 

728 

735 

757 

764 

772 

779 

786 

793 

801 

808 

830 

837 

844 

851 

Q59 

866 

873 

880 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

60 

N. 

60CT 

601 

602 

603 

604 

605 

606 

607 

608 

609 

610 

611 

612 

613 

614 

615 

616 

617 

618 

619 

620 

621 

622 

623 

624 

625 

626 

627 

628 

629 

630 

631 

632 

633 

634 

635 

636 

637 

638 

639 

640 

641 

642 

643 

644 

645 

646 

647 

648 

649 

650 

N. 


USEFUL  TABLES. 
Table— ( Continued). 


2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 

830 

837 

844 

851 

859 

866 

873 

880 

902 

909 

916 

924 

931 

938 

"945 

952 

974 

981 

988 

996 

#003 

#010 

#017 

*025 

046 

053 

061 

068 

075 

082 

089 

097 

118 

125 

132 

140 

117 

154 

161 

168 

190 

197 

204 

211 

219 

226 

233 

240 

262 

269 

276 

283 

290 

297 

305 

312 

8 

333 

340 

347 

355 

362 

369 

376 

383 

1 

0.8 

405 

412 

419 

426 

433 

440 

447 

455 

2 

1.6 

476 

483 

490 

497 

504 

512 

519 

526 

3 

2.4 

547 

554 

561 

569 

576 

583 

590 

“597 

4 

3.2 

618 

625 

633 

640 

647 

654 

661 

668 

5 

g 

4.0 
4 8 

689 

696 

704 

711 

718 

725 

732 

739 

7 

5.6 

760 

767 

774 

781 

789 

796 

803 

810 

g 

831 

838 

845 

852 

859 

866 

873 

880 

902 

909 

916 

923 

930 

937 

944 

951 

7.2 

972 

979 

986 

993 

#000 

#007 

*014 

*021 

043 

050 

057 

064 

071 

078 

085 

092 

113 

120 

127 

134 

141 

148 

155 

162 

183 

190 

197 

204 

211 

218 

225 

232 

253 

260 

267 

274 

281 

288 

295 

302 

323 

330 

337 

344 

"351 

“358 

"365 

“372 

7 

393 

400 

407 

414 

421 

428 

435 

442 

1 

0.7 

463 

470 

477 

484 

491 

498 

505 

511 

2 

1.4 

532 

539 

546 

553 

560 

567 

574 

581 

3 

2.1 

602 

609 

616 

623 

630 

637 

644 

650 

4 

2.8 

671 

678 

685 

692 

699 

706 

713 

720 

5 

3.5 

741 

748 

754 

761 

768 

775 

782 

789 

6 

4.2 

810 

817 

824 

831 

837 

844 

851 

858 

7 

4.9 

879 

886 

893 

900 

906 

913 

920 

927 

8 

5.6 

948 

955 

962 

969 

975 

982 

989 

996 

9 

6.8 

017 

024 

030 

037 

044 

-051 

058 

065 

085 

092 

099 

106 

113 

120 

127 

134 

154 

161 

168 

175 

182 

188 

195 

202 

223 

229 

236 

243 

250 

257 

264 

271 

291 

298 

305 

312 

318 

325 

332 

339 

359 

366 

373 

380 

387 

393 

400 

407 

6 

428 

434 

441 

448 

455 

462 

468 

475 

1 

0.6 

496 

502 

509 

516 

523 

530 

536 

513 

2 

1.2 

564 

570 

577 

584 

591 

598 

604 

611 

3 

1.8 

632 

638 

645 

652 

659 

665 

672 

679 

4 

2.4 

699 

706 

713 

720 

726 

733 

740 

747 

5 

g 

3.0 

3.6 

767 

774 

781 

787 

794 

801 

808 

814 

7 

4.2 

835 

841 

848 

855 

862 

868 

875 

882 

g 

4.8 

902 

909 

916 

922 

929 

936 

943 

949 

9 

5*4 

969 

976 

983 

990 

996 

#003 

*010 

*017 

037 

043 

050 

057 

064 

070 

077 

084 

104 

111 

117 

124 

131 

137 

144 

151 

171 

178 

184 

191 

198 

204 

211 

218 

238 

245 

251 

258 

265 

271 

278 

285 

305 

311 

318 

325 

331 

338 

345 

351 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

N. 

550 

651 

652 

653 

654 

655 

656 

657 

658 

659 

>60 

661 

662 

663 

664 

665 

666 

667 

668 

669 

170 

671 

672 

673 

674 

675 

676 

677 

678 

679 

80 

681 

682 

683 

684 

685 

686 

687 

688 

689 

90 

691 

692 

693 

694 

695 

696 

697 

698 

699 

00 

N. 


LOGARITHMS. 


61 


Table— ( Continued). 


L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

! 9 

F 

*.  P. 

81  291 

298 

305 

311 

318 

325 

331 

”338 

345 

351 

358 

365 

371 

378 

385 

391 

398 

405 

”4U 

“418 

425 

431 

438 

445 

451 

458 

465 

471 

478 

485 

491 

498 

505 

511 

518 

525 

531 

538 

544 

551 

•558 

564 

571 

578 

584 

591 

598 

604 

611 

617 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 

690 

697 

704 

710 

717 

723 

730 

737 

743 

750 

757 

763 

770 

776 

783 

790 

796 

803 

809 

816 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

954 

961 

968 

974 

981 

987 

994 

*000 

*007 

*014 

32  020 

027 

033 

040 

046 

053 

060 

066 

073 

i 079 

7 

086 

092 

099 

105 

112 

119 

125 

132 

138 

! 145 

l 

0.7 

151 

158 

164 

171 

178 

184 

191 

197 

1 204 

1 210 

2 

1 .4 

217 

223 

230 

236 

243 

249 

256 

' 263' 

269 

| 276 

3 

2.X 

282 

289 

295 

302 

308 

315 

321 

j 328' 

334 

1 341 

4 

2.8 

347 

354 

360 

367 

373 

580 

387 

393! 

400 

406 

5 

3.5 

413 

419 

426 

432 

439 

445 

452 

; 458 

465 

471 

6 

4.2 

478 

484 

491 

497 

504 

510 

517 

523! 

530 

536 

7 

4.9 

543 

549 

556 

562 

569 

575 

582 

: 588  j 

595 

601 

8 

9 

5.6 

6.3 

607 

614 

620 

627 

633 

640 

646 

653  j 

659 

666 

672 

679 

685 

692 

698 

705 

711 

; 718  i 

”m 

^30 

737| 

743 

750 

756 

763 

769 

! 776 

' 782 : 

789 

795 

802, 

808 

814 

821 

827 

834 

840 ! 

! 847: 

853 

860 

866 

872 

■ 879! 

885 

898 

905 1 

1 911 

918 

924 

930 

837; 

943 

950 

956 

963 

969! 

975' 

982 

988 

995 1*001  *008  *014 

*020 

*027 

*033  *040' 

*046  *052 

33  0591 

065 

072 

078 

085 

091 

097, 

104! 

110 

117 

123 

129 

136 

142 

149 

155! 

161: 

168 

174 

181 

187: 

193 

200 

o 

213 

219, 

225 1 

232 

238 

245 

251 1 

257, 

264  i 

270. 

276 

283  ( 

289 

296 

302 

308 

315 

321 

327 

334 

340 

347 

353 1 

359 

366 

372 

6 

378 

385  i 

391 

398 

404 

410 

417J 

423 

429, 

436 

1 

0.6 

. 442 

448 

455 

461 1 

467 

474 

480' 

487 

493, 

499 

2 

1.2 

506: 

512 

518 

525! 

531 

537 

544, 

5501 

556 

563 

3 

1.8 

569| 

575; 

582 

588 

594 

601 

607 1 

613; 

620 

626 

4 

2.4 

632i 

639  i 

645 

651  i 

658 

664 

670 

677! 

683 

689 

5 

3.0 

696 

702 

708 

715 

721 

727 

734! 

740 ! 

746 

753 

6 

3.6 

759 

765 

771] 

778, 

784 

790 

797 1 

803 ; 

809 

816 

7 

4.2 

822 

828 

835 

841  j 

847 

853 

860  j 

866 

872, 

879 

8 

4.8 

885 

”891 

“897 

"904j 

910 

916 

”923 

”929 ; 

”935! 

942 

9 

5.4 

948 

"954 

”960 

“967! 

”973 

”979 

“985 

“992 

”998, 

*004 

34  Oil 

017 

023 

029! 

036 

042 

048 

055 

061 

067 

073 

080 

086 

092 

098 

105 

111 

117 

123 

130 

136 

142 

148 

155 

161 

167 

173 

180 

186 

192 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

261 

267 

273 

280 

286 

292 

298 

305 

311 

317 

323 

330 

336 

342 

348 

354 

361 

367 

373 

379 

386 

392 

398 

404 

410 

417 

423 

429 

435 

442 

448 

454 

460 

466 

473 

479 

485 

491 

497 

504 

510 

516 

522 

528 

535 

541 

547 

553 

559 

566 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 

62 

N. 

70CT 

701 

702 

703 

704 

705 

706 

707 

708 

709 

710 

711 

712 

713 

714 

715 

716 

717 

718 

719 

720 

721 

722 

723 

724 

725 

726 

727 

728 

729 

730 

731 

732 

733 

734 

735 

736 

737 

738 

739 

740 

741 

742 

743 

744 

745 

746 

747 

748 

749 

750 

N. 


USEFUL  TABLES. 


Table — ( Continued). 


2 

3 

4 

5 

6 

7 

8 

9 

P 

>.  P. 

522 

528 

535 

541 

547 

553 

559 

566 

584 

590 

597 

603 

609 

615 

“621 

"628 

646 

652 

658 

665 

671 

677 

683 

689 

708 

714 

720 

726 

733 

739 

745 

751 

770 

776 

782 

788 

794 

800 

HOT 

813 

831 

837 

844 

850 

856 

862 

868 

874 

893 

899 

905 

911 

917 

924 

930 

936 

7 

954 

960 

967 

973 

979 

985 

991 

997 

l 

0.7 

016 

022 

028 

034 

040 

046 

052 

058 

2 

1.4 

077 

083 

089 

095 

101 

107 

114 

120 

3 

2.1 

138 

144 

150 

156 

163 

169 

175 

181 

4 

2.8 

199 

205 

211 

217 

224 

230 

236 

242 

5 

g 

3.5 

4.2 

260 

266 

272 

278 

285 

291 

297 

303 

7 

4.9 

321 

327 

333 

339 

345 

352 

358 

364 

g 

5.6 

382 

388 

394 

400 

406 

412 

418 

425 

9 

443 

449 

455 

461 

467 

473 

479 

485 

503 

509 

516 

522 

528 

534 

540 

546 

564 

570 

576 

582 

r.s< 

594 

600 

606 

625 

631 

637 

643 

649 

655 

661 

667 

685 

691 

697 

703 

709 

715 

721 

727 

745 

751 

757 

763 

769 

775 

781 

788 

806 

812 

818 

824 

830 

836 

842 

848 

6 

866 

872 

878 

884 

890 

896 

902 

908 

1 

0.6 

926 

932 

938 

944 

950 

956 

962 

968 

2 

1.2 

986 

992 

998 

*004 

*010 

*016 

*022 

*028 

3 

1.8 

046 

052 

058 

064 

070 

076 

082 

088 

4 

2.4 

106 

112 

118 

124 

130 

136 

141 

147 

5 

3.0 

165 

171 

177 

183 

189 

195 

201 

207 

6 

3.6 

225 

231 

237 

243 

249 

255 

261 

267 

7 

4.2 

285 

291 

297 

303 

308 

314 

320 

326 

8 

4.8 

e a 

344 

350 

356 

362 

368 

374 

380 

386 

9 

0.4 

404 

410 

415 

421 

427 

433 

~439 

“445 

463 

469 

475 

481 

487 

493 

499 

504 

522 

528 

534 

540 

546 

552 

558 

564 

581 

587 

593 

599 

605 

611 

617 

623 

641 

646 

652 

658 

664 

670 

676 

682 

700 

705 

711 

717 

723 

729 

735 

741 

5 

759 

764 

770 

776 

782 

788 

794 

800 

1 

0.5 

817 

823 

829 

835 

841 

847 

853 

859 

2 

1.0 

876 

882 

888 

894 

900 

906 

911 

917 

3 

1.5 

“935 

“941 

"947 

“953 

“958 

964 

“970 

“976 

4 

2.0 
O R 

~994 

999 

*005 

*011 

*017 

*023 

*029 

*035 

5 

g 

2.5 

3.0 

052 

058 

064 

070 

075 

081 

087 

093 

7 

3.5' 

111 

116 

122 

128 

134 

140 

146 

151 

g 

4.0 

169 

175 

181 

186 

192 

198 

204 

210 

9 

4.5 

227 

233 

239 

245 

251 

256 

262 

268 

286 

291 

297 

303 

309 

315 

320 

326 

344 

349 

355 

361 

367 

373 

379 

384 

402 

408 

413 

419 

425 

431 

437 

442 

460 

466 

371 

477 

483 

489 

495 

500 

518 

“523 

529 

535 

“541 

“5  + 7 

“552 

"558 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 

. 

50 

751 

752 

753 

754 

755 

756 

757 

758 

759 

60 

761 

762 

763 

764 

765 

766 

767 

768 

769 

70 

771 

772 

773 

774 

775 

776 

777 

778 

779 

80 

781 

782 

783 

784 

785 

786 

787 

788 

789 

90 

791 

792 

793 

794 

795 

796 

797 

798 

799 

too 

N. 


LOGARITHMS. 


63 


Table— ( Continued). 


2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

518 

523 

529 

535 

541 

547 

552 

558 

576 

581 

587 

593 

599 

604 

610 

• ill, 

633 

639 

645 

651 

656 

662 

668 

674 

691 

697 

703 

708 

714 

.720 

726 

731 

749 

754 

760 

766 

772 

777 

783 

789 

806 

812 

818 

823 

829 

835 

841 

846 

864 

869 

875 

881 

887 

892 

898 

904 

921 

927 

933 

938 

944 

950 

955 

961 

978 

984 

990 

996 

*001 

*007 

*013 

*018 

036 

Oil 

047 

053 

058 

064 

070 

076 

093 

098 

104 

110 

116 

~121 

127 

133 

150 

156 

161 

167 

173 

178 

184 

190 

6 

207 

213 

218 

224 

230 

235 

241 

247 

l 

0.6 

264 

270 

275 

281 

287 

292 

298 

304 

2 

1.2 

321 

326 

332 

338 

343 

349 

355 

360 

3 

1.8 

377 

383 

389 

395 

400 

406 

412 

417 

4 

2.4 

434 

440 

446 

451 

457 

463 

468 

474 

5 

3.0 

491 

497 

502 

508 

513 

519 

525 

530 

6 

3.6 

547 

553 

559 

564 

570 

576 

5*1 

587 

7 

4.2 

604 

610 

615 

621 

627 

632 

638 

643 

8 

4.8 

9 

5.4 

660 

666 

672 

677 

683 

689 

694 

700 

717 

722 

728 

734 

739 

"745 

"750 

"756 

773 

779 

784 

790 

795 

801 

807 

812 

829 

835 

840 

846 

852 

857 

863 

885 

891 

897 

902 

908 

913 

919 

925 

941 

947 

953 

958 

964 

969 

975 

981 

997 

*003 

*009 

*014 

*020 

*025 

*031 

*037 

053 

059 

064 

070 

076 

081 

087 

092 

109 

115 

120 

126 

131 

137 

143 

148 

165 

170 

176 

182 

187 

193 

198 

204 

"m 

226 

232 

"237 

243 

"248 

"254 

"260 

"276 

282 

"287 

"293 

~298 

“304 

"310 

315 

5 

332 

337 

343 

348 

354 

360 

365 

371 

1 

0.5 

387 

393 

39* 

404 

409 

415 

421 

426 

2 

1.0 

443 

448 

454 

459 

465 

470 

476 

481 

3 

1.5 

498 

504 

509 

515 

520 

526 

531 

537 

4 

2.0 

553 

559 

564 

570 

575 

581 

586 

592 

5 

2.5 

609 

614 

620 

625 

631 

636 

642 

647 

6 

3.0 

664 

669 

675 

680 

686 

691 

697 

702 

7 

3.5 

719 

724 

730 

735 

741 

746 

752 

757 

8 

4.0 

774 

779 

785 

790 

796 

801 

807 

812 

9 

4.5 

"829 

"834 

840 

"845 

851 

"856 

“862 

867 

883 

889 

894 

900 

905 

911 

916 

922 

938 

944 

949 

955 

960 

966 

971 

977 

993 

998 

*004 

*009 

*015 

*020 

*026 

*031 

048 

053 

059 

064 

069 

075 

080 

086 

102 

108 

113 

119 

124 

129 

135 

140 

157 

162 

168 

173 

179 

184 

189 

195 

211 

217 

222 

227 

233 

238 

244 

249 

266 

271 

276 

282 

287 

293 

298 

304 

320 

"325 

331 

"336 

"342 

“347 

"352 

"358 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

64 


USEFUL  TABLES. 


Table— ( Continued). 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

I 

\ P. 

800 

90  309 

314 

320 

325 

331 

336 

“342 

"347 

352 

358 

801 

363 

369 

274 

380 

385 

390 

396 

401 

407 

412 

802 

417 

423 

428 

434 

439 

445 

450 

455 

461 

466 

803 

472 

477 

482 

488 

493 

499 

504 

509 

515 

520 

804 

526 

531 

536 

542 

547 

553 

558 

563 

569 

574 

805 

580 

585 

590 

596 

601 

607 

612 

617 

623 

628 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

807 

687 

693 

698 

703 

709 

714 

720 

725 

730 

736 

808 

741 

747 

752 

757 

763 

768 

773 

779 

784 

789 

809 

795 

800 

806 

811 

816 

822 

827 

832 

838 

843 

810 

849 

854 

859 

865 

870 

875 

"881 

“886 

891 

"897 

811 

902 

907 

913 

918 

924 

929 

“934 

940 

“945 

950 

1 

6 

812 

956 

961 

966 

972 

977 

982 

988 

993 

998 

*004 

0.6 

813 

91  009 

014 

020 

025 

030 

036 

041 

046 

052 

057 

2 

1.2 

814 

062 

068 

073 

078 

084 

089 

094 

100 

105 

110 

3 

1.8 

815 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

4 

2.4 

816 

169 

174 

180 

185 

190 

196 

201 

200 

212 

217 

5 

3.0 

817 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

6 

3.6 

818 

275 

281 

286 

291 

297 

302 

307 

312 

318 

323 

7 

4.2 

819 

328 

334 

339 

344 

350 

355 

360 

365 

371 

376 

8 

9 

4.8 

5.4 

820 

381 

387 

392 

397 

403 

408 

"413 

“418 

424 

429 

821 

434 

440 

445 

450 

'455 

461 

*466 

*471 

“477 

“482 

822 

487 

492 

498 

503 

508 

514 

519 

524 

529 

535 

823 

540 

545 

551 

556 

561 

566 

572 

577 

582 

587 

824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

' 

825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

826 

698 

703 

709 

714 

719 

724 

730 

735 

740 

745 

827 

751 

756 

761 

766 

772 

777 

782 

787 

793 

798 

828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

829 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

830 

908 

“913 

"918 

924 

“929 

”934 

“939 

"944 

“950 

"955 

831 

960 

"965 

“971 

~976 

“981 

"986 

"991 

"997 

*002 

•HI)  7 

5 

832 

92  012 

018 

023 

028 

033 

038 

044 

049 

054 

059 

1 

0.5 

833 

065 

070 

075 

080 

085 

091 

096 

101 

106 

111 

2 

1.0 

834 

117 

122 

127 

132 

137 

143 

148 

153 

158 

163 

3 

1.5 

835 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

4 

2.0 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

5 

2.5 

837 

273 

278 

283 

288 

293 

298 

304 

309 

314 

319 

6 

3.0 

838 

324 

330 

335 

340 

345 

350 

355 

361 

366 

371 

7 

3.5 

839 

376 

381 

387 

392 

397 

402 

407 

412 

418 

423 

8 

4.0 

840 

428 

"433 

~443 

"449 

“454 

“459 

“464 

469 

“474 

9 

4.5 

841 

480 

”485 

"490 

”495 

“500 

“505 

"5ll 

“516 

“521 

“526 

842 

531 

536 

542 

547 

552 

557 

562 

567 

572 

578 

843 

583 

588 

593 

59s 

603 

609 

614 

619 

624 

629 

844 

634 

639 

645 

650 

655 

660 

665 

670 

675 

681 

845 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 

846 

737 

742 

747 

752 

758 

763 

768 

773 

77s 

783 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

334 

848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 

849 

891 

996 

901 

906 

911 

916 

921 

927 

932 

937 

850 

942 

“947 

“952 

957 

“962 

“967 

“973 

978 

“983 

“988 

N. 

L.O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 

. 

i50 

851 

852 

853 

854 

855 

856 

857 

858 

859 

60 

861 

862 

863 

864 

865 

866 

867 

868 

869 

70 

871 

872 

f«73 

874 

875 

876 

877 

878 

879 

80 

881 

882 

883 

884 

885 

886 

887 

888 

889 

90 

891 

892 

893 

894 

895 

896 

897 

898 

899 

00 

N. 


LOGARITHMS. 


65 


Table— ( Continued). 


2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

952 

957 

962 

967 

973 

978 

983 

988 

*003 

*008 

*013 

*018 

*024 

*029 

*034 

*039 

054 

059 

064 

069 

075 

080 

085 

090 

105 

110 

115 

120 

125 

131 

136 

141 

156 

161 

166 

171 

176 

181 

186 

192 

207 

212 

217 

222 

227 

232 

237 

242 

258 

263 

268 

273 

278 

283 

288 

293 

6 

308 

313 

318 

323 

328 

33 1 

339 

344 

1 

0.6 

359 

364 

369 

374 

379 

384 

389 

394 

2 

1.2 

409 

4-4 

420 

425 

430 

435 

440 

445 

3 

1.8 

460 

465 

470 

4 75 

480 

485 

490 

495 

4 

2.4 

510 

515 

520 

526 

”53l 

"536 

"541 

546 

5 

6 

3.0 

3.6 

561 

566 

571 

576 

581 

586 

591 

596 

7 

4.2 

611 

616 

621 

626 

631 

636 

641 

646 

8 

4.8 

661 

666 

671 

676 

682 

687 

892 

897 

9 

5.4 

712 

717 

722 

727 

732 

737 

742 

747 

762 

767 

772 

777 

782 

787 

792 

797 

812 

817 

822 

827 

832 

837 

842 

847 

862 

867 

872 

877 

882 

887 

892 

897 

912 

917 

922 

927 

932 

937 

942 

947 

962 

967 

972 

977 

982 

987 

992 

”997 

012 

017 

022 

027 

~032 

"037 

"042 

"047 

5 

062 

067 

072 

077 

082 

086 

091 

096 

1 

0.5 

111 

116 

121 

126 

131 

136 

141 

146 

2 

1.0 

161 

166 

171 

176 

181 

186 

191 

196 

3 

1.5 

211 

216 

221 

226 

231 

236 

240 

245 

4 

2.0 

260 

265 

270 

275 

280 

285 

290 

295 

5 

2.5 

310 

315 

320 

325 

330 

335 

340 

345 

6 

3.0 

359 

364 

369 

374 

379 

384 

389 

394 

7 

3.5 

409 

414 

419 

424 

429 

433 

438 

443 

8 

4.0 

"458 

"463 

"468 

"473 

"478 

~483 

”488 

^93 

9 

4.5 

507 

"512 

"517 

“522 

"527 

~532 

*537 

542 

557 

562 

567 

571 

576 

581 

586 

591 

606 

611 

616 

621 

626 

630 

635 

640 

655 

660 

665 

670 

675 

680 

685 

689 

704 

709 

714 

719 

724 

729 

734 

738 

753 

758 

763 

768 

773 

778 

783 

787 

4 

802 

807 

812 

817 

822 

827 

832 

836 

1 

0.4 

851 

856 

861 

866 

871 

876 

880 

885 

2 

0.8 

900 

905 

910 

915 

919 

924 

929 

934 

3 

1.2 

"949 

"954 

"959 

"963 

"968 

"973 

"978 

”983 

4 

1.6 

998 

*002 

*007 

*012 

*0l7 

*022 

*027 

■■■032 

5 

g 

2.0 

2.4 

D46 

051 

056 

061 

066 

071 

075 

080 

7 

2.8 

095 

100 

105 

109 

114 

119 

124 

129 

8 

3.2 

143 

148 

153 

158 

163 

168 

173 

177 

9 

3.6 

192 

197 

202 

207 

211 

216 

221 

226 

240 

245 

250 

255 

260 

265 

270 

274 

289 

294 

299 

303 

308 

313 

318 

323 

337 

342 

347 

352 

357 

361 

366 

371 

386 

390 

395 

-400 

405 

410 

415 

419 

434 

439 

444 

448 

453 1 458 

463 

468 

2 

3 

4 

5 

6 

1 7 

8 

9 

P 

. P. 

66 


USEFUL  TABLES. 


Table— ( Continued). 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

900 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

901 

472 

477 

482 

487 

492 

497 

501 

506 

511 

516 

902 

521 

525 

530 

535 

540 

545 

550 

554 

559 

564 

903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

905 

665 

670 

674 

679 

684 

6.-9 

694 

698 

703 

708 

906 

713 

718 

722 

727 

732 

737 

742 

746 

751 

756 

907 

761 

766 

770 

775 

780 

785 

789 

794 

799 

804 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

942 

”947 

911 

952 

957 

961 

966 

971 

976 

“980 

“985 

”990 

995 

5 

912 

999 

*004 

*009 

*014 

*019 

*023 

*028 

*033 

*038 

*042 

1 

0.5 

913 

96  047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

2 

1.0 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

3 

1.5 

915 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

4 

2.0 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

5 

2.5 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

6 

3.0 

918 

284 

289 

294 

298 

303 

308 

313 

317 

322 

327 

7 

3.5 

919 

332 

336 

341 

346 

350 

355 

360 

365 

369 

374 

8 

9 

4.0 

920 

379 

384 

388 

393 

398 

402 

407 

”412 

"417 

”421 

4.5 

921 

426 

431 

435 

440 

445 

'450 

”454 

”459 

464 

“468 

922 

473 

478 

483 

487 

492 

497 

501 

506 

511 

515 

923 

520 

525 

530 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

581 

586 

591 

595 

600 

605 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

703 

927 

708 

713 

717 

722 

727 

731 

736 

741 

745 

750 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

"853 

858 

“862 

"867 

”872 

”876 

”881 

”886 

”890 

931 

895 

"900 

"904 

“909 

914 

”918 

“923 

”928 

“932 

”937 

4 

- 932 

942 

946 

951 

956 

960 

965 

970 

974 

979 

984 

1 

0.4 

933 

988 

993 

997 

*002 

*007 

*011 

*016 

*021 

*025 

’030 

2 

0.8 

934 

97  035 

039 

044 

049 

053 

058 

063 

067 

072 

077 

3 

1.2 

935 

081 

086 

090 

095 

100 

104 

109 

114 

118 

123 

4 

1.6 

936 

128 

132 

137 

142 

146 

151 

155 

160 

165 

169 

5 

2.0 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

6 

2.4 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

7 

2.8 

939 

267 

271 

276 

280 

285 

290 

294 

299 

304 

308 

8 

3.2 

940 

313 

~317 

322 

“327 

"331 

”336 

“340 

”345 

”350 

”354 

9 

3.6 

941 

359 

“364 

”368 

373 

“377 

“382 

“387 

391 

”396 

”400 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

943 

451 

456 

460 

465 

470 

474 

479 

483 

488 

493 

944 

497 

502 

506 

511 

516 

520 

525 

529 

534 

539 

945 

543 

548 

552 

557 

562 

566 

571 

575 

580 

585 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

948 

681 

685 

690 

695 

699 

704 

708 

713 

717 

722 

949 

727 

731 

736 

740 

745 

749 

754 

759 

763 

768 

950 

772 

777 

782 

786 

791 

795 

800 

804 

809 

813 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P.  P. 

LOGARITHMS. 


67 


T able— ( Continued ) . 


N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 I 

8 

9 

P 

. P. 

950 

97  772 

~777 

782 

“786 

"791 

705 

800 

“804 

N 111 

“813 

951 

818 

823 

827 

832 

836 

841 

*845 

“850 

”855 

859 

952 

864 

868 

873 

877 

882 

886 

891 

896! 

900 

905 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

950 

954 

955 

959 

964 

868 

973 

978 

982 

987 

991 

996 

955 

98  000 

005 

009 

014 

019 

023 

028 

032 

037 

041 

956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

957 

091 

096 

100 

105 

109 

114 

118 

123 

127 

132 

958 

137 

141 

146 

150 

155 

159 

164 

168 

173 

177 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

“254 

“259 

"263 

"268 

961 

272 

277 

281 

286 

290 

295 

"299 

“304 

"308 

“313 

5 

962 

318 

322 

327 

331 

336 

340 

345 

349 1 

354 

358 

1 

0.5 

963 

363 

367 

372 

376 

381 

385 

390 1 

394; 

399 

403 

2 

1.0 

964 

408 

412 

417 

421 

426 

430 

435  j 

439! 

444 

448 

3 

1.5 

965 

453 

457 

462 

466 

471 

475 

4801 

484  j 

489 

493 

4 

2.0 

966 

498 

502 

507 

511 

516 

520 

525 

529i 

534 

538 

5 

2 .5 

967 

543 

547 

552 

556 

561 

565 

570 

574| 

579 

583 

3.0 

968 

588 

592 

597 

601 

605 

610 

614 

l 619! 

623 

628 

7 

3.5 

969 

632 

637 

641 

646 

650 

655 

659 

i 664; 

668 

673 

8 

9 

4.0 

4.5 

970 

677 

682 

686 

691 

695 

700, 

"704 

1 709 1 

Tl3 

“717 

971 

722 

726 

731 

735 

740 

744 

“749 

!“753l 

"758 

“762 

972 

7671 

771 

776 

780 

784 

789 

793 

i 798 

802 

807 

973 

811! 

816 

820 

825 

829 

834 

838 

! 843! 

847 

851 

974 

8561 

860 

865 

869 

874 

878 

883 

1 887; 

892 

896 

975 

900 

905 

i 909 

914 

918 

923 

927 

932  i 

936 

941 

976 

945  i 

949 

954 

958 

963 

967 

972 

1 976 1 

981 

985 

977 

9891 

j 994 

998 

*003 

*007 

*012 

*016 

*021 

*025 

*029 

978 

99  034 

038 

043 

1 047 

j 052 

056 

1 061 

! 065 

069 

074 

979 

078 

1 083 

087 

092 

! 096 

100 

105 

j 109 

I 114 

118 

980 

123 

hm 

Hi 

136 

"lio 

145 

"149 

154 

“l58 

~162 

981 

167 

171 

176 

180 

185 

189 

193 

i 198 

1 202 

207 

4 

982 

211 

216 

220 

224 

229 

233 

238 

'242 

247 

251 

1 

0.4 

983 

255 

260 

264 

269 

273 

277 

282 

! 286 

291 

295 

2 

0.8 

984 

300 

304 

308 

313 

317 

322 

326 

1 330 

335 

339 

3 

1.2 

985 

344 

348 

352 

357 

361 

366 

370 

374 

379 

383 

4 

1.6 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

5 

2.0 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

471 

6 

2.4 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

7 

2.8 

989 

520 

524 

528 

533 

537 

542 

546 

550 

555 

559 

8 

3.2 

990 

564 

“568 

“572 

“577 

“581 

"585 

"590 

“594 

"599 

”603 

9 

3.6 

991 

607 

“612 

“616 

"621 

"625 

“629 

“634 

“638 

“642 

“647 

992 

651 

656 

660 

664 

669 

673 

677 

682 

i 686 

691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

730 

734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

! 774 

778 

995 

782 

787 

791 

795 

800 

804 

808 

813 

i 817 

822 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

997 

870 

874 

878 

.883 

887 

891 

896 

900 

904 

909 

998 

913 

917 

922 

926 

930 

935 

939 

944 

948 

952 

999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

1000 

00  000 

004 

“009 

1)13 

"017 

“022 

“026 

“030 

“035 

”039 

N. 

L.  0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

P 

. P. 

68 


USEFUL  TABLES. 


TRIGONOMETRIC  FUNCTIONS.. 


DIRECTIONS  FOR  USING  THE  TABLE. 

The  table  given  on  pages  74-78  contains  the  natural  sines, 
cosines,  tangents,  and  cotangents  of  angles  from  0°  to  90°. 
Angles  less  than  45°  are  given  in  the  first  column  at  the  left- 
hand  side  of  the  page,  and  the  names  of  the  functions  are 
given  at  the  top  of  the  page;  angles  greater  than  45°  appear 
at  the  right-hand  side  of  the  page,  and  the  names  of  the  func- 
tions are  given  at  the  bottom.  Thus,  the  second  column  con- 
tains the  sines  of  angles  less  than  45°  and  the  cosines  of  angles 
greater  than  45°;  the  sixth  column  contains  the  cotangents  of 
angles  less  than  45°  and  the  tangents  of  angles  greater  than 
45°.  To  find  the  function  of  an  angle  less  than  45°,  look  in 
the  column  of  angles  at  the  left  of  the  page  for  the  angle,  and 
at  the  top  of  the  page  for  the  name  of  the  function;  to  find 
a function  of  an  angle  greater  than  45°,  look  in  the  column 
at  the  right  of  the  page  for  the  angle  and  at  the  bottom  of  the 
page  for  the  name  of  the  function.  The  successive  angles 
differ  by  an  interval  of  10';  they  increase  downwards  in  the 
left-hand  column  and  upwards  in  the  right-hand  column. 
Thus,  for  angles  less  than  45°  read  down  from  top  of  page,  and 
for  angles  greater  than  45°  read  up  from  bottom  of  page. 

The  third,  fifth,  seventh,  and  ninth  columns,  headed  d , 
contain  the  differences  between  the  successive  functions;  for 
example,  in  the  second  column  we  find  that  the  sine  of  32°  10' 
is  .5324  and  that  the  sine  of  32°  20'  is  .5348;  the  difference  is 
.5348  —.5324  = .0024,  and  the  24  is  written  in  the  third  column, 
just  opposite  the  space  between  .5324  and  .5348.  In  like  man- 
ner the  differences  between  the  successive  tabular  values  of 
the  tangents  are  given  in  the  fifth  column,  those  between  the 
cotangents  in  the  seventh  column,  and  those  for  the  cosines 
in  the  ninth  column.  These  differences  in  the  functions  cor- 
respond to  a difference  of  10'  in  the  angle;  thus,  when  the 
angle  32°  10'  is  increased  by  10',  that  is,  to  32°  20',  the  increase 
of  the  sine  is  .0024,  or,  as  given  in  the  table,  24.  It  will  be 
observed  that  in  the  tabular  difference  no  attention  is  paid  to 
the  decimal  point,  it  being  understood  that  the  difference  is 


TRIGONOMETRIC  FUNCTIONS. 


G9 


merely  the  number  obtained  by  subtracting  the  last  two  or 
three  figures  of  the  smaller  function  from  those  of  the  larger. 
These  differences  are  used  to  obtain  the  sines,  cosines,  etc. 
of  angles  not  given  in  the  table;  the  method  employed  may 
be  illustrated  by  an  example.  Required,  the  tangent  of 
27°  34'.  Looking  in  the  table,  we  see  that  the  tangent  of  27°  30' 
is  .5206,  and  (in  column  5)  the  difference  for  10'  is  37.  Differ- 
ence for  1'  is  37  -f- 10  = 3.7,  and  difference  for  4'  is  3.7  X 4 = 14.8. 
Adding  this  difference  to  the  value  of  the  tan  27°  30',  we  have 
tan  27°  30'  = .5206 
difference  for  4'  = 14.8 

tan  27°  34'  = .5220.8  or  .5221,  to  four  places. 

Since  only  four  decimal  places  are  retained,  the  8 in  the 
fifth  place  is  dropped  and  the  figure  in  the  fourth  place  is 
increased  by  1,  because  8 is  greater  than  5. 

To  avoid  multiplication,  the  column  of  proportional  parts, 
headed  P.  P.,  at  the  extreme  right  of  the  page,  is  used.  At 
the  head  of  each  table  in  this  column  is  the  difference  for  10', 
and  below  are  the  differences  for  any  intermediate  number 
of  minutes  from  1'  to  9'.  In  the  above  example,  the  differ- 
ence for  10'  was  37;  looking  in  the  table  with  37  at  the  head, 
the  difference  opposite  4 is  14.8;  that  opposite  7 is  25.9;  and  so 
on.  For  want  of  space,  the  differences  for  the  cotangents  for 
angles  less  than  45°  (or  the  tangents  of  angles  greater  than 
45°)  have  been  omitted  from  the  tables  of  proportional  parts. 
The  use  of  these  functions  should  be  avoided,  if  possible, 
since  the  differences  change  very  rapidly,  and  the  computa- 
tion is  therefore  likely  to  be  inexact.  The  method  to  be 
employed  when  dealing  with  these  functions  may  be  shown 
by  an  example:  Required,  the  tangent  of  76°  34'.  Since  this 
angle  is  greater  than  45°,  wTe  look  for  it  in  the  column  at  the 
right,  and  read  up;  opposite  the  76°  30',  we  find,  in  sixth  col- 
umn, the  number  4.1653,  and  corresponding  to  it  in  seventh 
column  is  the  difference  540.  Since  540  is  the  difference  for  10', 
the  difference  for  4'  is  540  X ^ = 216.  Adding  this  difference: 
tan  76°  30'  = 4.1653 
difference  for  4'  = 216 


tan  76°  34'  = 4.1869 


70 


USEFUL  TABLES. 


When  the  angle  contains  a certain  number  of  seconds, 
divide  the  number  by  6,  and  take  the  whole  number  nearest 
to  the  quotient;  look  out  this  number  in  the  table  of  propor- 
tional parts  (under  the  proper  difference ),  and  take  out  the 
number  that  is  opposite  to  it.  Shift  the  decimal  point  one 
place  to  the  left,  and  then  add  it  to  the  partial  function 
already  found. 

Find  the  sine  of  34°  26'  44". 

sine  34°  2 O'  = .5640 
difference  for  6'  = 14.4 

difference  for  44"  = 1.7 


Difference  for  10'  = 24. 


sine  34°  26'  44"  = .5656 


-b*  = 7}.  Look  out  in  the  P.  P. 
table  the  number  under  24 
and  opposite  7.  It  is  16.8. 
Shifting  the  decimal  point 
one  place  to  the  left,  we  get 
1.68,  or,  say,  1.7. 

The  tangent  is  found  in  the  same  way  as  the  sine. 


To  find  the  cosine  of  an  angle: 

As  the  angle  increases,  the  value  of  the  cosine  decreases, 
so  that,  instead  of  adding  the  values  corresponding  to  6'  and 
44"  to  the  function  already  found,  we  subtract  them  from  it. 
Thus,  find  cos  34°  26'  44". 

cos  34°  20'  = .8258  Difference  for  10'  = 17. 


difference  for  6'  = 10.2 

difference  for  44"  = 1.2 

total  difference  = 11.4 

.8247 


The  number  under  the  17  and 
opposite  the  7,  in  the  P.  P. 
table,  is  11.9.  Therefore,  take 
1.19,  or,  say,  1.2. 


Therefore,  cos  34°  26'  44"  = .8258  - .0011  = .824°. 

Only  four  decimal  places  are  kept;  therefore,  the  figure  of 
the  difference  following  the  decimal  point  is  dropped  before 
subtracting. 

The  cotangent  is  found  in  the  same  manner. 

We  will  now  consider  angles  greater  than  45°. 

Find  the  sine  of  68°  47'  22". 

In  obtaining  the  difference , it  must  be  remembered  to 
choose  the  one  between  the  sine  of  68u  40'  and  the  next  angle 
above  it,  namely,  68°  50'. 


TRIGONOMETRIC  FUNCTIONS. 


71 


sine  68°  40'  = .9315 
difference  for  7'  = 7 

difference  for  19"  = .4 


Difference  for  10'  = 10. 


sine  68°  47'  22"  = .9322 


* = 3f,  say  4.  Under  the  10 
and  opposite  the  4 is  the 
number  4.0;  shifting  the  deci- 
mal point,  we  get  .4. 

As  usual,  only  four  decimal  places  are  kept. 

The  tangent  is  found  in  the  same  manner. 

Find  cos  68°  47'  22". 

As  before,  the  cosine  decreases  as  the  angle  increases; 
therefore,  we  subtract  the  successive  sine  values  correspond- 
ing to  the  increments  in  the  angle.. 


cos  68°  40'  = .3638 
difference  for  7'  = 18.9 

difference  for  22"  = 1.1 

total  difference  = 


Difference  for  lO'  = 27. 


20 

.3618 


Under  the  27  and  opposite  the 
4 is  the  number  10.8;  there- 
fore, take  1.08  in  this  case, 
or,  say,  1.1. 

Therefore,  cos  68°  47'  22"  = .3638  - .002  = .3618. 

The  cotangent  is  found  in  the  same  way. 

In  finding  the  functions  of  an  angle,  the  only  difficulty 
likely  to  be  encountered  is  to  determine  whether  the  differ- 
ence obtained  from  the  table  of  proportional  parts  is  to  be 
added  or  subtracted.  This  can  be  told  in  every  case  by 
observing  whether  the  function  is  increasing  or  decreasing  as 
the  angle  increases.  For  example,  take  the  angle  21°;  its 
sine  is  .3584,  and  the  following  sines,  reading  downwards, 
are  .3611,  .3638,  etc.  It  is  plain,  therefore,  that  the  sine  of 
say  21°  6'  is  greater  than  that  of  21°,  and  that  the  difference 
for  6'  must  be  added.  On  the  other  hand,  the  cosine  of  21° 
is  .9336,  and  the  following  cosines,  reading  downwards,  are 
.9325,  .9315,  etc.;  that  is,  as  the  angle  grows  larger  the  cosine 
decreases.  The  cosine  of  an  angle  between  21°  and  21°  10', 
say  21°  6',  must  therefore  lie  between  .9325  and  .9315;  that  is, 
it  must  be  smaller  than  .9325,  which  shows  that  in  this  case 
the  difference  for  6'  must  be  subtracted  from  the  cosine  of  21°. 

We  will  now  consider  the  case  in  which  the  function,  i.  e., 
the  sine,  cosine,  tangent,  or  cotangent,  is  given  and  the  cor- 
responding angle  is  to  be  found. 


72 


USEFUL  TABLES. 


Find  the  angle  whose  sine  is  .4943.  The  operation  is 
arranged  as  follows: 


.4943 

Difference  for 

.4924 

= sin  29°  30'. 

1st  remainder  19 

18.2 

= difference  for  7'. 

2d  remainder  .8 

.78  = difference  for  .3'  or  18". 

.4943  = sin  29°  37'  18". 

Looking  down  the  second  column,  we  find  the  sine  next 
smaller  than  .4943  to  be  .4924,  and  the  difference  for  10'  to 
be  26.  The  angle  corresponding  to  .4924  is  29°  30'.  Sub- 
tracting the  .4924  from  .4943,  the  first  remainder  is  19;  looking 
in  the  table  of  proportional  parts,  the  part  next  lower  than 
this  difference  is  18.2,  opposite  which  is  7'.  Subtracting  this 
difference  from  the  remainder,  we  get  .8,  and,  looking  in  the 
table,  we  see  that  7.8  with  its  decimal  point  moved  one  place 
to  the  left  is  nearest  to  the  second  difference.  This  is  the 
difference  for  .3'  or  18".  Hence,  the  angle  is  29°  30'  + 7' 
■f  18  = 29°  37'  18". 

Find  the  angle  whose  tangent  is  .8824. 


.8824 

Difference  for  10'  = 51. 

.8796 

= tan  41°  20'. 

1st  remainder  28 

25.5 

= difference  for  5'. 

2d  remainder  2.5 

2.55  = difference  for  .5'  or  30". 

.8824  = tan  41°  25'  30". 

In  the  two  examples  just  given,  the  minutes  and  seconds 
corresponding  to  the  1st  and  2d  remainders  are  added  to  the 
angle  taken  from  the  table.  Thus,  in  the  first  example,  an 
inspection  of  the  table  shows  that  the  angle  increases  as  the 
sine  increases;  hence,  the  angle  whose  sine  is  .4943  must  be 
greater  than  29°  30',  whose  sine  is  .4924.  For  this  reason  the 
correction  must  be  added  to  29°  30'.  The  same  reasoning 
applies  to  the  second  example. 


TRIGONOMETRIC  FUNCTIONS. 


73 


Find  the  angle  whose  cosine  is  .7742. 

.7742  Difference  for  10'  = 18. 

.7735  = cos  39°  20'. 

1st  remainder  7 

5.4  = difference  for  3'. 

2d  remainder  1.6 

1.62  = difference  for  .9'  or  54". 

39°  20'  — 3'  54"  = 39°  16'  6",  which  is  the  angle  whose 
cosine  is  .7742. 

Looking  down  the  eighth  column,  headed  cos,  the  next 
smaller  cosine  is  .7735,  to  which  corresponds  the  angle 
39°  20'.  The  difference  for  10'  is  18.  Subtracting,  the  remain- 
der is  7,  and  the  next  lower  number  in  the  table  of  propor- 
tional parts  is  5.4,  which  is  the  difference  for  3'.  Subtracting 
this  from  1st  remainder,  2d  remainder  is  1.6,  which  is  nearest 
16.2  of  table  of  proportional  parts,  if  the  decimal  point  of  the 
latter  is  moved  to  the  left  one  place.  Since  16.2  corresponds 
to  a difference  of  9',  1.62  corresponds  to  a difference  of  .9% 
or  54".  Hence,  the  correction  for  the  angle  39°  20'  is  3'  54", 
From  the  table,  it  appears  that,  as  the  cosine  increases,  the 
angle  grows  smaller;  therefore,  the  angle  whose  cosine  is  .7742 
must  be  smaller  than  the  angle  whose  cosine  is  .7735,  and  the 
correction  for  the  angle  must  be  subtracted. 

Find  the  angle  whose  cotangent  is  .9847. 


.9847 

Difference  for  10'  = 57. 

.9827 

= cos  45°  30'. 

1st  remainder  20 

17.1 

= difference  for  3'. 

2d  remainder  2.9 

2.85  = difference  for  .5'  or  30". 

45°  30'  — 3'  30"  = 45°  26'  30",  the  angle  whose  cotangent 
is  .9847. 

In  finding  the  angle  corresponding  to  a function,  as  in 
the  above  examples,  the  angles  obtained  may  vary  from  the 
true  angle  by  2 or  3 seconds;  in  order  to  obtain  the  number 
of  seconds  accurately,  the  functions  should  contain  six  or 
seven  decimal  places. 


o 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 


).0058 

).0087 

).0116 

).0145 

>.0175 

>.0204 

).0233 

).0262 

).0291 

).0320 

).0349 

>76378 

).0407 

).0437 

).0466 

).0495 

).0524 

).0553 

).0582 

>.0612 

>.0641 

>.0670 

>.0699 

>.0729 

>.0758 

>.0787 

1.0816 

1.0846 

>.0875 

76904 

1.0934 

>.0963 

1.0992 

(.1022 

(.1051 

U080 

(.1110 

.1139 

.1169 

.1198 

0228 

0257" 

1287 

1317 

.1346 

0376 

.1405 

.1435 

.1465 

.1495 

.1524 

0554 

0584 


.1257 


Cot. 


infinit. 

343.7737 

171.8854 

114.5887 

85.9398 

68.7501 

57.2900 

49.1039 

42.9641 

38.1885 

34.3678 

31.2416 

28.6363 

26.4316 

24.5418 

22.9038 

21.4704 

20.2056 

19.0811 

18.0750 

17.1693 

16.3499 

15.6048 

14.9244 

14.3007 

13.7267 

13.1969 

12.7062 

12.2505 

11.8262 

11.4301 

11.0594 

10.7119 

10.3854 

10.0780 

9.7882 

9.5144 

9.2553 

9.0098 

8.7769 

8.5555 

8.3450 

~~  8.1443 

7.9530 

7.7704 

7.5958 

7.4287 

7.2687 

7.1154 

6.9682 

6.8269 

6.6912 

6.5606 

6.4348 

81861 

61398 

47756 

38207 

31262 

26053 

22047 

18898 

16380 

14334 

12648 

11245 

10061 

905’ 

8194 

7451 

6804 

6237 

5740 

5298 

4907 

4557 

4243 

3961 

3707 

3475 

3265 

3074 

2898 

2738 

2591 

2455 

2329 

2214 

2105 

2007 

1913 

1826 

1746 

1671 

1600 

1533 

1472 

1413 

1357 

1306 

1258 

1210 


Cot.  d.  Tan.  I d. 


Cos. 

1.0000 

1.0000 

.0000 

1.0000 

0.9999 

0.9999 

0.9998 

0.9998 

0.9997 

0.9997 

0.9996 

0.9995 

0.9994 

0.9993 

0.9992 

0.9990 

0.9989 

0.9988 

0.9986 

0.9985 

0.9983 

0.9981 

0.9980 

0.9978 

09976 

679974 

0.9971 

0.9969 

0.9967 

0.9964 

09962 

679959 

0.9957 

0.9954 

0.9951 

0.9948 


0.9945 

09942 

0.9939 

0.9936 

0.9932 

0.9929 

09925 


0.9922 

0.9918 

0.9914 

0.9911 

0.9907 


0.9903 


P 

. P. 

<>  90 

50 

30 

40 

30 

1 

3.0 

20 

2 

6.0 

10 

3: 

9.0 

0 89 

4' 

5 

12.0 

15.0 

50 

6 

18.0 

40 

7 

21.0 

30 

8 

24.0 

20 

10 

0 88 

9 

27.0 

29 

50 

•1 

; 2.9 

40 

2 

j 5.8 

30 

3 

| 8.T 

20 

4 

! 11.6 

10 

5 

' 14.5 

0 87 

6 

17.4 

50 

40 

30 

7 

8 

1 20.3 
! 23.2 

9 

; 26.1 

20 

10 

0 86 

1 

28 

2.8 

50 

2 

5.6 

40 

3 

8.4 

30 

4 

11.2 

20 

5 

14.0 

10 

6 

16.8 

0 85 

7 

8 

19.6 

22.4 

50 

9 

25.2 

40 

30 

20 

5 

10 

1 

0.5 

0 84 

2 

1.0 

1.5 

2.0 

50 

4 

40 

5 

2.5 

30 

6 

3.0 

20 

7 

3.5 

10 

8 

4.0 

0 83 

9 

4.5 

50 

40 

4 

30 

20 

1 

2 

0.4 

0.8 

10 

3 

1.2 

o 82 

4 

1.6 

50 

5 

2.0 

40 

6 

2.4 

30 

7 

2.8 

8 

3.2 

— 

9 

3.6 

t o 

P 

. P. 

0 

10 

•20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 


,32 


Cot. 
6.3138 
6.1970 
6.0844 
5.9758 
5.8708 
5.7694 
5.6713 
5 .57 64 
5.4845 
5.3955 
5.3093 
5.2257 
5.1446 
5^0658 
4.9894 
4.9152 
4.8430 
4.7729 
4/7046 
4.6382 
4.5736 
4.5107 
4.4494 
4.3897 
4.3315 
4.2747 
4.2193 
4.1653 
4.1126 
4.0611 
4.0108 


3.9617 

3.9136 

3.8667 

3.8208 

3.7760 


3.7321 

jT.6891 

3.6470 

3.6059 

3.5656 

3.5261 

3.4874 

3^4495 

3.4124 

3.3759 

3.3402 

3.3052 

3.2709 

3.2371 

3.2041 

3.1716 

3.1397 

3.1084 

3.0777 


Cot.  d.  Tan.  d.  Sin.  d. 


Cos. 

0.9877 

0.9872 

0.9868 

0.9863 

0.9858 

0.9853 


0.9848 


0.9793 

0.9787 


0.9696 

0.9689 

0.9681 

0.9674 

0.9667 

0.9659 

(L9652 

0.9644 

0.9636 

0.9628 

0.9621 

0^9613 

0.9605 

0.9596 

0.9588 

0.9580 

0.9572 


0.9563 


0.9555 

0.9546 

0.9537 

0.9528 

0.9520 


0.9511 


0 81 

50 


0 80 

50 


10 

0 79 

50 


0 78 


10 

077 

50 

40 

30 

20 

10 

0 76 

50 

40 

30 

20 

10 

0 75 

50 


o 74 


0 73 


o 72 


P, 

. P. 

32 

31 

30 

1 

3.2 

3.1 

3.0 

2 

6.4 

6.2 

6.0 

3 

9.6 

9.:'. 

9.0 

4 

12  8 

12.4 

12.0 

5 

16.0 

15.5 

15.0 

6 

19.2 

18.6 

18.0 

7 

22.4 

21.7 

21.0 

8 

24.8 

24.0 

9 

28.8 

27.9 

27.0 

29 

28 

27 

1 

2.9 

2.8 

2.7 

2 

5.8 

5.6 

5.4 

3 

8.7 

8.4 

8.1 

4 

11.6 

11.2 

10.8 

5 

14.5 

14.0 

13.5 

6 

17.4 

16.8 

16.2 

7 

20.3 

19.6 

18.9 

8 

23.2 

22.4 

21.6 

9 

26.1 

25.2 

24.3 

9 8 

0.9  0.8 

1.8 1 1.6 

2.72.4 

3.63.2 
4.5  4.0 

5.4  4.8 
6.3 1 5.6 

7.2  6.4 

8.1 1 7.2 


7 6 

1 0.7  0.6 

2 1.41.2 

3 2.11.8 

4 2.8j  2.4 

5 3.5  3.0 

6 4.23.6 

7 4.9  4.2 

8 5.6  4.8 

9 6.3i5.4 


5 I’ 4 

1 0.50.4 

2 1 .0 ! 0.8 

3 1.51.2 

4 2.0 ! 1.6 

5 2.5'2.0 

6 3.0  2.4 

7 3.5 j 2.8 

8 4.0  3.2 

9 4.5, 3.6 


P.  P. 


/ 

~0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 

10 

20 

30 

40 

50 

0 


Tan. 


0.5095 


Cot.  d. 


Cot. 


3.0777 

3.0475 

3.0178 

2.9887 

2.9600 

2.9319 

2.9042 

2.8770 

2.8502 

2.8239 

2.7980 

2.7725 

2.7475 

2.7228 

2.6985 

2.6746 

2.6511 

2.6279 

2.6051 

2.5826 

2.5605 

2.5386 

2.5172 

2.4960 

2.4751 

2.4545 

2.4342 

2.4142 

2.3945 

2.3750 

2.3559 

2.3369 

2.3183 

2.2998 

2.2817 

2.2637 

2.2460 

2.2286 

2.2113 

2.1943 

2.1775 

2.1609 

2.1445 

2.1283 

2.1123 

2.0965 

2.0809 

2.0655 

2.0503 

2.0353 

2.0204 

2.0057 

1.9912 

1.9768 

1.9626 

Tan. 


Cos. 

0.9511 


0.9502 

0.9492 

0.9483 

0.9474 

0.9465 


0.9455 


0.9*46 

0.9436 

0.9426 

0.9417 

0.940' 


0.9397 


0.9387 

0.9377 

0.9367 

0.9356 

0.9346 


0.9336 


0.9325 

0.9315 

0.9304 

0.9293 

0.9283 


0.9272 


0.9261 

0.9250 

0.9239 

0.9228 

0.9216 


0.9205 


0.9194 

0.9182 

0.9171 

0.9159 

0.9147 


0.9135 


0.9124 

0.9112 

0.9100 

0.9088 

0.9075 


•0.9063 


0.9051 

0.9038 

0.9026 

0.9013 

0.9001 


0.8988 


0.8975 

0.8962 

0.8949 

0.8936 

0.8923 

0.8910 


d.  Sin.  d. 


o 72 

50 


o 71 


o 70 

50 


0 69 


0 67 


0 66 


0 65 

50 


0 64 

50 


o 63 


I’. 

P. 

37 

36 

35 

1 

3.7 

3.6 

3.5 

2 

7.4 

7.2 

7.0 

3 

11.1 

10.8 

10.5 

4 

14.8 

14.4 

14.0 

5 

18.5 

18.0 

17.5 

6 

22.2 

21.6 

21.0 

7 

25.9 

25.2 

24.5 

8 

29.6 

28.8 

'28.0 

9 

33.3,32.4 

.31.5 

34 

33 

32 

1 

3.4 

3.3 

3.2 

2 

6.8 

6.6 

6.4 

3 

10.2 

9.9 

9.6 

4 

13.6 

13.2 

12.8 

5 

17.0 

16.5 

16.0 

6 

20.4 

19.8 

19.2 

7 

23.8 

23.1 

[22.4 

8 

27.2 

26.4  25.6 

9 

30.6 

29.7128.8 

28 

27 

26 

1 

2.8 

2.7 

2.6 

2 

5.6 

5.4 

5.2 

3 

8.4 

8.1 

7.8 

4 

11.2 

10.8 

10.4 

5 

14.0 

13.5 

13.0 

6 

16.8 

16.2 

15.6 

7 

19.6 

18.9 

18.2 

8 

22.4 

21.6 

20.8 

9 

25.2, 

,24.3,23.4 

10.4  9.6 
11.7;10.8 


II 


I0|  9 

1.0j0.9 
2.0  1.8 
3.02.7 

4.0  3.6 
5.04.5 

6.0  j 5.4 

7.0  6.3 

8.0  7.2 
9.9,9.08.1 


P.  P. 


p.  p. 

\ 

44 

143 

CM 

1 

4.4 

4.3 

4.2 

2 

8.8 

8.6 

8.4 

3 

13.2 

12.9 

12.6 

4 

17.6 

17.2 

16.8 

5 

22.0  21.5  21.0 

, 6 

26.4 

25.8  25.2 

5 7 

30.8|30.1 

29.4 

8 

35.2  34.4 ! 33.6 

9 

39.6,38.7,37.8 

41 

40 

39 

1 

4.1 

4.0 

3.9 

2 

8.2 

8.0 

7.8 

1 3 

12.3 

12.0 

11.7 

4 

16.4 

16.0 

15.6 

5 

20.5 

20.0 

19.5 

6 

24.6 

24.0 

23.4 

7 

28.7 

28.0 

27.3 

8 

32.8 

32.0 

31.2 

9 

36.9 

36.0 

35.1 

38 

3.8 

7.6 

11.4 

15.2 
19.0 
22.8 
26.6 

30.4 

34.2 


37 

3.7 

7.4 

11.1 

14.8 

18.5 

22.2 

25.9 

29.6 
33.3 


26 

25 

24 

1 

2.6 

2.5 

2.4 

2 

5.2 

5.0 

4.8 

3 

7.8 

7.5 

7.2 

4 

10.4 

10.0 

9.6 

4 

13.0 

12.5 

12.0 

6 

15.6 

15.0 

14.4 

7 

18.2 

17.5 

16.8 

8 

20.8 

20.0 

19.2 

9 

23.4 

22.5 

21.6 

23 

17 

16 

1 

2.3 

1.7 

1.6 

2 

4.6 

3.4 

3.2 

3 

6.9 

5.1 

4.8 

4 

9.2 

6.8 

6.4 

5 

11..') 

8.5 

8.0 

6 

13.S 

10.2 

9.6 

7 

16.1 

11.9 

11.2 

8 

18.4 

13.6 

12.8 

9 

20.7 

15.3 

14.4 

15 

14 

13 

1 

1.5 

1.4 

1.3 

2 

3.0 

2.8 

2.6 

3 

4.5 

4.2 

3.9 

4 

6.0 

5.6 

5.2 

5 

7.5 

7.0 

6.5 

6 

9.0 

8.4 

7.8 

7 

10.5 

9.8 

9.1 

8 

12.0 

11.2 

10.4 

9 

13.5 

12.6 

11.7 

P. 

P. 

Sin. 

j 0.5878 
0.5901 
) 0.5925  ; o 
) 0.5948  24 


- 23 


) 0.5972 
) 0.5995 


Tan. 

0.7265 

0.7310 

0.7355 

0.7400 

0.7445 

0.7490 

0.7536 

0.7581 

0.7627 

0.7673 

0.7720 

0.7766 

0.7813 

0.7860 

0.7907 

0.7954 

0.8002 

o.8o;o 

0.8098 

0.8146 

0.8195 

0.8243 

0.8292 

0.8342 

0.8391 

0.8441 

0.8491 

0.8541 

0.8591 

0.8642 

0.8693 

0-8744 

0.8796 

0.8847 

0.8899 

0.8952 

0.9004 

Cos.  d. 


0.9057 
0.9110  53 
0.9163  54 
0.9217 
0.9271 

54 

55 


0.9325 


0.9380 
0.943  j 
0.9490 
0.9545 
0.9601 


0.9657 


0.9713 

0.9770 

0.9827 


Cot. 

1.3764 

1.3680 

1.3597 

1.3514 

1.3432 

1.3351 

1.3270 

1.3190 

1.3111 

1.3032 

1.2954 

1.2876 

1.2799 

1.2723 
1.2647 
1.2572 
1 .2497 
1.2423 

1.2349 

1.2276 

1.2203 

1.2131 

1.2059 

1.1988 

1.1918 

1.1847 

1.1778 

1.1708 

1.1640 

1.1571 

1.1504 

1.1436 
1 .1369 
1.1303 
1.1237 
1.1171 

1.1106 

1.1041 

1.0977 

1.0913 

1.0850 

1.0786 

1.0724 

1.0661 
1 .0599 
1 .0538 
1.0477 
1.0416 

1.0355 

1.0295 

1.0235 

1.0176 

1.0117 

1.0058 

1.0000 

, Tan. 

Cos.  i 

108090 

0.8073 

0.8056 

0.8039 

0.8021 

0.8004 

0.7986 

0.7969 

0.7951 

0.7934 

0.7916 

0.7898 

0.7880 

0.7862 

0.7844 

0.7826 

0.7808 

0.7790 

0.7771 

0.7753 

0.7735 

0.7716 

0.7698 

0.7679 

0.7660 

0.7642 

0.7623 

0.7604 

0.7585 

0.7566 

0.7547 

! 0.7528 
; 0.7509 
; 0.7490 
, 0.7470 
0.7451 

0.7431 

. 0.7412 
: 0.7392 
i 0.7373 
. 0.7353 
’ 0.7333 

; 0.7314 

! 0.7294 
' 0.7274 
0.7254 
0.7234 
0.7214 

0.7193 

0.717:! 
, 0.7153 
, 0.7133 
, 0.7112 
0.7092 

1 0.7071 

d.  Sin.  d.  ' ° 


0 54 

50 


0 53 


0 52 

50 


o 51 


0 50 


0 49 

50 


0 48 

50 


10 

0 47 

50 


0 46 

50 


0 45 


P.  P. 


58 

5.8 

11.6 

17.4 

23.2 
29.0 
34.8 
40.6 

46.4 

52.2 

54 

5.4 


10.8 

16.2 

21.6 

27.0 

32.4 

37.8 

43.2 

48.6 

50 


55 

5.5 

11.0 

16.5 

22.0 

27.5 

33.0 

38.5 

44.0 

49.5 
51 
5.1 


10.2 

15.3 

20.4 


10.4  1 
15.6  1 
20.8  2 
26.5  26.0  25.5 
31.8:31.2  30.6 
37.1  36.4  35.7 
42.4! 4 1.6  40.8 
47.7,46.8  45.9 

49  1 48 

4.9  4.8 
9.8 l 9.6 
14.714.4 
19.6  19.2 


5.0 


24.0 

28.8 

33.6 

38.4 

43.2 

45 


14.1 
18.8 

23.5  23.0  22.5 

28.2  27 .6  j 27 .0 
32.9  32.2  31.5 

37.6  36.8'36.0 

42.3  41.4] 40.5 


24 

23 

22 

21 

1 

2.4 

2.3 

2.2 

2.1 

2 

4.8 

4.6 

4.4 

4.2 

3 

7.2 

' 6.9 

6.6 

6.3 

4 

9.6 

9.2 

8.8 

8.4 

5 

12.0 

11.5 

11.0 

10.5 

6 

14.4 

13.8 

13.2 

12.6 

7 

16.8 

16.1 

15.4 

14.7 

8 

19.2 

18.4 

17.6 

16.8 

9 

21.6 

20.7 

19.8 

18.9 

20 

19 

18 

17 

1 

2.0 

1.9 

1.8 

1.7 

2 

4.0 

3.8 

3.6 

3.4 

3 

6.0 

5.7 

5.4 

5.1 

4 

8.0 

7.6 

7.2 

6.8 

5 

10.0 

9.5 

9.0 

8.5 

6 

12.0 

11.4 

10.8 

10.2 

7 

14.0 

13.3 

12.6 

11.9 

8 

16.0 

15.2 

14.4 

13.6 

9 

18.0 

17.1 

|l6.2 

15.3 

P.  P. 


PRIME  NUMBERS. 


79 


PRIME  NUMBERS. 

Every  prime  number  is  an  odd  number  and  has  for  its  unit 
figure  1,  3,  7,  or  9;  any  odd  number  that  has  5 for  its  unit  fig- 
ure is  divisible  by  5,  and  is  not  a prime  number.  The  prime 
factors  of  any  number  less  than  1,000  may  be  found  from  the 
following  table.  If  the  number  is  odd  and  does  not  end  with 
5,  the  factors  are  given  directly;  thus,  the  prime  factors  of 
357  are  3,  7,  and  17;  those  of  931  are  7, 7,  and  19,  the  exponent 
2 of  the  7 indicating  that  7 is  used  twice  as  a factor.  If  a 
number  is  a prime  number,  the  space  beside  it  is  blank;  thus, 
317  and  859  are  prime  numbers.  To  find  the  prime  factors  of 
an  odd  number  that  has  5 for  the  unit  figure,  divide  by  5 
until  a quotient  is  obtained  which  does  not  have  5 for  a unit 
figure;  the  factors  of  this  quotient  are  then  found  from  the 
table,  and  with  the  5’s  already  used  as  divisors  constitute 
the  prime  factors.  For  example,  to  find  the  prime  factors  of 
5,775  proceed  as  follows:  5,775-4-5  = 1,155;  1,155-4-5  = 231; 
from  the  table,  231  = 3 X 7 X 11;  hence,  5,775  = 3 X 5 X 5 X 
7 X 11.  If  the  number  is  even,  divide  it  by  2,  the  quotient  by 
2,  and  so  on  until  an  odd  quotient  is  reached;  then  find  the 
prime  factors  of  the  quotient  from  the  table.  The  process  of 
finding  the  prime  factors  of  936  is  as  follows: 

936  -4-  2 = 468;  468  -4-  2 = 234;  234  2 = 117;  117  = 32  X 13, 

from  table.  Hence,  936  = 2'  X 32  X 13  = 2 X 2 X 2 X 3 X 3 X 13. 


FACTORS  OF  3.1416. 


Not  Regarding  Decimal  Point,  3.1416  = 


2 X 15708 

22  X 1428 

68  X 462 

3 X 10472 

24  X 1309 

77  X 408 

4 X 7854 

28  X 1122 

84  X 374 

6 X 5236 

33  X 952 

88  X 357 

7 X 4488 

34  X 924 

102  X 308 

8 X 3927 

42  X 748 

119  X 264 

11  X 2856 

44  X 714 

132  X 238 

12  X 2618 

51  X 616 

136  X 231 

14  X 2244 

56  X 561 

154  X 204 

17  X 1848 
21  X 1496 

66  X 476 

168  X 187 

BO 


USEFUL  TABLES. 


PRIME  FACTORS. 

Prime  Factors  of  All  Odd  Numbers  From  1 to  1,000 
That  Are  Not  Divisible  by  5. 


1 

101 

201 

3-67 

301 

7-43 

401 

3 

103 

203 

7-29 

303 

3-101 

403 

13-31 

7 

107 

207 

3-23 

307 

407 

11-37 

9 

32 

109 

209 

11-19 

309 

3-103 

409 

11 

111 

3-37 

211 

311 

411 

3-137 

13 

113 

213 

3-71 

313 

413 

7-59 

17 

117 

32T3 

217 

7-31 

317 

417 

3-139 

19 

119 

7.17 

219 

3-73 

319 

11-29 

419 

21 

3-7 

121 

112 

221 

13T7 

321 

3-107 

421 

23 

123 

3-41 

223 

323 

17-19 

423 

32-47 

27 

33 

127 

227 

327 

3-109 

427 

7-61 

29 

129 

3-43 

229 

329 

7-47 

429 

3T1-13 

31 

131 

231 

3-7-11 

331 

431 

33 

3T1 

133 

7-19 

233 

333 

32-37 

433 

37 

137 

237 

3-79 

337 

437 

19-23 

39 

3T3 

139 

239 

339 

3T13 

439 

41 

141 

3-47 

241 

341 

'11-31 

441 

32.72 

43 

143 

11-13 

243 

35 

343 

73 

443 

47 

147 

3-72 

247 

13-19 

347 

447 

3-149 

49 

72 

149 

249 

3-83 

349 

449 

51 

3T7 

151 

251 

351 

33-13 

451 

11-41 

53 

153 

32-17 

253 

11-23 

353 

453 

3151 

57 

319 

157 

257 

357 

3-7T7 

457 

59 

159 

3-53 

259 

7-37 

359 

459 

33-17 

61 

161 

7-23 

261 

32-29 

361 

192 

461 

63 

32.7 

163 

263 

363 

3-112 

463 

67 

167 

267 

3-89 

367 

467 

69 

3*23 

169 

132 

269 

369 

32-41 

469 

7-67 

71 

171 

32'19 

271 

371 

7-53 

471 

3-157 

73 

173 

273 

3-7T3 

373 

473 

11-43 

77 

7*11 

177 

3-59 

277 

377 

13-29 

477 

32-53 

79 

179 

279 

32-31 

379 

479 

81 

34 

181 

281 

381 

3-127 

481 

13-37 

83 

183 

3-61 

283 

383 

483 

3-7-23 

87 

3-29 

187 

11-17 

287 

7-41 

387 

32-43 

487 

89 

189 

33-7 

289 

172 

389 

489 

3-163 

91 

7-13 

191 

291 

3-97 

391 

17-23 

491 

93 

3*31 

193 

293 

393 

3-131 

493 

17-29 

97 

197 

297 

33-U 

397 

497 

7-71 

99 

32-11 

199 

299 

13-23 

399 

3-7T9 

499 

PRIME  FACTORS. 


81 


Prime  Factors  of  All  Odd  Numbers  From  1 to  1,000 
That  Are  Not  Divisible  by  5. 

( Continued ). 


501 

3167 

601 

701 

801 

32-89 

901 

17-53 

503 

603 

32'67 

703 

19-37 

803 

11-73 

903 

3-7-43 

507 

3-132 

607 

707 

7-101 

807 

3-269 

907 

509 

609 

3-7-29 

709 

809 

909 

32-101 

511 

7-73 

611 

13-47 

711 

32-79 

811 

911 

513 

33T9 

613 

713 

23-31 

813 

3-271 

913 

11-83 

517 

11-47 

617 

717 

3-239 

817 

19-43 

917 

7131 

519 

3-173 

619 

719 

819 

32-7-13 

919 

521 

621 

33‘23 

721 

7-103 

821 

921 

3-307 

523 

623 

7-89 

723 

3-241 

823 

923 

13-71 

527 

17-31 

627 

3T1T9 

727 

827 

927 

32-103 

529 

232 

629 

17-37 

729 

3e 

829 

929 

531 

32'59 

631 

731 

17-43 

831 

3-277 

931 

72-19 

533 

13-41 

633 

3-211 

733 

833 

72-17 

933 

3-311 

537 

3-179 

637 

72-13 

737 

11-67 

837 

33"31 

937 

539 

72T1 

639 

32-71 

739 

839 

939 

3-313 

541 

641 

741 

3-13-19 

841 

292 

941 

543 

3181 

643 

743 

843 

3-281 

943 

23-41 

547 

647 

747 

32-83 

847 

7-112 

947 

549 

32*61 

649 

11-59 

749 

7-107 

849 

3-283 

949 

13-73 

551 

19-29 

651 

3-7-31 

751 

851 

23-37 

951 

3-317 

553 

7-79 

653 

753 

3-251 

853 

953 

557 

657 

32-73 

757 

857 

957 

3-11-29 

559 

13-43 

659 

759 

3-11-23 

859 

959 

7137 

561 

3-11-17 

661 

761 

861 

3-7-41 

961 

312 

563 

663 

3T3-17 

763 

7*109 

863 

963 

32-107 

567 

34*7 

667 

23-29 

767 

13-59 

867 

3 -172 

967 

569 

669 

3-223 

769 

869 

11-79 

969 

3T7-19 

571 

671 

11-61 

771 

3-257 

871 

13-67 

971 

573 

3-191 

673 

773 

873 

32-97 

973 

7-139 

577 

677 

777 

3-7-37 

877 

977 

579 

3-193 

679 

7.97 

779 

19-41 

879 

3-293 

979 

11-89 

581 

7-83 

681 

3-227 

781 

11-71 

881 

981 

32-109 

583 

11-53 

683 

783 

33‘29 

883 

983 

587 

687 

3-229 

787 

887 

987 

3-7-47 

589 

19-31 

689 

13-53 

789 

3-263 

889 

7T27 

989 

23-43 

591 

3-197 

691 

791 

7T13 

891 

34"11 

991 

593 

693 

32-7-11 

793 

13*61 

893 

19-47 

993 

3-331 

597 

3-199 

697 

17-41 

797 

897 

3-13*23 

997 

599 

699 

3-233 

799 

17-47 

899 

29-31 

999 

33*37 

82 


USEFUL  TABLES. 


CIRCUMFERENCES  AND  AREAS  OF 
CIRCLES  FROM  1-64  TO  100. 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

.0491 

.0002 

4% 

13.7445 

15.0330 

.0982 

.0008 

4K 

14.1372 

15.9043 

.1963 

.0031 

4% 

14.5299 

16.8002 

Vs 

.3927 

.0123 

4/4 

4J| 

14.9226 

17.7206 

.5890 

.0276 

15.3153 

18.6555 

li 

.7854 

.0491 

5 

15.7080 

19.6350 

£ 

.9817 

.0767 

5% 

16.1007 

20.6290 

% 

1.1781 

.1104 

5M 

16.4934 

21.6476 

£ 

1.3744 

.1503 

5/4 

16.8861 

22.6907 

% 

1.5708 

.1963 

534 

17.2788 

23.7583 

£ 

1.7671 

.2485 

5^4 

17.6715 

24.8505 

1.9635 

.3068 

5^4 

18.0642 

25.9673 

£ 

2.1598 

.3712 

5% 

18.4569 

27.1086 

2.3562 

.4418 

6 

18.8496 

28.2744 

2.5525 

.5185 

6/4 

19.2423 

29.4648 

Pa 

2.7489 

.6013 

6 M 

19.6350 

30.6797 

if 

2.9452 

.6903 

(% 

20.0277 

31.9191 

1 

3.1416 

.7854 

634 

20.4204 

33.1831 

lVs 

3.5343 

.9940 

6% 

20.8131 

84.4717 

3.9270 

1.2272 

6/4 

21.2058 

35.7848 

1% 

4.3197 

1.4849 

6j| 

21.5985 

37.1224 

4.7124 

1.7671 

7 

21.9912 

38.4846 

1% 

5.1051 

2.0739 

734 

22.3839 

39.8713 

1/4 

1% 

5.4978 

2.4053 

7)4 

22.7766 

41.2826 

5.8905 

2.7612 

23.1693 

42.7184 

2 

6.2832 

3.1416 

7 )f 

23.5620 

44.1787 

2% 

6.6759 

3.5466 

75/| 

23.9547 

45.6636 

2 y* 

7.0686 

3.9761 

7 % 

m 

24.3474 

47.1731 

2ya 

7.4613 

4.4301 

24.7401 

48.7071 

7.8540 

4.9087 

8 

25.1328 

50.2656 

2&s 

8.2467 

5.4119 

8)4 

25.5255 

51.8487 

8.6394 

5.9396 

834 

25.9182 

53.4563 

2,4 

9.0321 

6.4918 

8/4 

26.3109 

55.0884 

3 

9.4248 

7.0686 

8)4 

26.7036 

56.7451 

334 

9.8175 

7.6699 

f/a 

27.0963 

58.4264 

334 

10.2102 

8.2958 

8% 

27.4890 

60.1322 

Wa 

10.6029 

8.9462 

8/4 

27.8817 

61.8625 

33 4 

10.9956 

9.6211 

9 

28.2744 

63.6174 

3% 

11.3883 

10.3206 

9)4 

934 

28.6671 

65.3968 

3/4 

11.7810 

11.0447 

29.0598 

67.2008 

3% 

12.1737 

11.7933 

9% 

29.4525 

69.0293 

4 

12.5664 

12.5664 

9)4 

29.8452 

70.8823 

434 

12.9591 

13.3641 

9*1 

30.2379 

72.7599 

434 

13.3518 

14.1863 

tfi 

30.6306 

74.6621 

TABLE  OF  CIRCLES. 


83 


Table— ( Continued ) . 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

9% 

31.0233 

76.589 

15% 

49.0875 

191.748 

10 

31.4160 

78.540 

15% 

49.4802 

194.828 

10% 

31.8087 

80.516 

15% 

49.8729 

197.933 

iok 

32.2014 

82.516 

16 

50.2656 

201.062 

10% 

32.5941 

84.541 

16% 

50.6583 

204.216 

io% 

32.9868 

86.590 

16% 

51.0510 

207.395 

10% 

33.3795 

88.664 

1&\ 

51.4437 

210.598 

10% 

33.7722 

90.763 

16% 

51.8364 

213.825 

10% 

34.1649 

92.886 

16% 

52.2291 

217.077 

11 

34.5576 

95.033 

16% 

52.6218 

220.354 

11  % 

34.9503 

97.205 

16% 

53.0145 

223.655 

11 % 

35.3430 

99.402 

17 

53.4072 

226.981 

ii% 

35.7357 

101.623 

17% 

53.7999 

230.331 

n% 

36.1284 

103.869 

17% 

54.1926 

233.706 

n% 

36.5211 

106.139 

17% 

54.5853 

237.105 

n% 

36.9138 

108.434 

17% 

54.9780 

240.529 

11% 

37.3065 

110.754 

17% 

55.3707 

243.977 

12 

37.6992 

113.098 

17% 

55.7634 

247.450 

12% 

38.0919 

115.466 

17% 

56.1561 

250.948 

12% 

38.4846 

117.859 

18 

56.5488 

254.470 

12% 

38.8773 

120.277 

18% 

56.9415 

258.016 

12% 

39.2700 

122.719 

18% 

57.3342 

261.587 

12% 

39.6627 

125.185 

18% 

57.7269 

265.183 

12% 

40.0554 

127.677 

18% 

58.1196 

268.803 

12% 

40.4481 

130.192 

18% 

58.5123 

272.448 

13 

40.8408 

132.733 

18% 

58.9050 

276.117 

13% 

41.2335 

135.297 

18% 

59.2977 

279.811 

13% 

41.6262 

137.887 

19 

59.6904 

283.529 

13% 

42.0189 

140.501 

19% 

60.0831 

287.272 

3-3% 

42.4116 

143.139 

19% 

60.4758 

291.040 

13% 

42.8043 

145.802 

19% 

60.8685 

294.832 

13  % 

43.1970 

148.490 

19% 

61.2612 

298.648 

13% 

43.5897 

151.202 

19% 

61.6539 

302.489 

14 

43.9824 

153.938 

19% 

62.0466 

306.355 

14% 

44.3751 

156.700 

19% 

62.4393 

310.245 

14% 

44.7678 

159.485 

20 

62.8320 

314.160 

14% 

45.1605 

162.296 

20% 

63.2247 

318.099 

1434 

45.5532 

165.130 

20% 

63.6174 

322.063 

3-4% 

45.9459 

167.990 

20% 

64.0101 

326.051 

14% 

46.3386 

170.874 

20% 

64.4028 

330.064 

14% 

46.7313 

173.782 

20% 

64.7955 

334.102 

15 

47.1240 

176.715 

20% 

65.1882 

338.164 

15% 

47.5167 

179.673 

20% 

65.5809 

342.250 

15% 

47.9094 

182.655 

21 

65.9736 

346.361 

15% 

48.3021 

185.661 

21% 

66.3663 

350.497 

15% 

48.6948 

188.692 

21% 

66.7590 

354.657 

84 


USEFUL  TABLES. 


Table — ( Continued). 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

21% 

67.1517 

358.842 

27% 

85.2159 

577.870 

91 1/ 

67.5444 

363.051 

27% 

85.6086 

583.209 

21^| 

67.9371 

367.285  • 

27% 

86.0013 

588.571 

21% 

68.3298 

371.543 

27% 

86.3940 

593.959 

21% 

68.7225 

375.826 

27fl 

86.7867 

599.371 

22 

69.1152 

380.134 

27% 

87.1794 

604.807 

22% 

69.5079 

384.466 

27% 

87.5721 

610.268 

2234 

69.9006 

388.822 

28 

87.9648 

615.754 

70.2933 

393.203 

28% 

88.3575 

621.264 

2234 

70.6860 

397.609 

28% 

88.7502 

626.798 

2g| 

71.0787 

402.038 

28% 

89.1429 

632.357 

71.4714 

406.494 

28% 

89.5356 

637.941 

22^ 

71.8641 

410.973 

28% 

89.9283 

643.549 

23 

72.2568 

415.477 

28% 

90.3210 

649.182 

23% 

72.6495 

420.004 

28% 

90.7137 

654.840 

93% 

73.0422 

424.558 

29 

91.1064 

660.521 

23|| 

73.4349 

429.135 

29% 

91.4991 

666.228 

73.8276 

433.737 

29% 

91.8918 

671.959 

23% 

74.2203 

438.364 

29% 

92.2845 

677.714 

23% 

74.6130 

443.015 

29% 

92.6772 

683.494 

23% 

75.0057 

447.690 

29^| 

93.0699 

689.299 

24 

75.3984 

452.390 

29% 

93.4626 

695.128 

24% 

75.7911 

457.115 

29% 

93.8553 

700.982 

24% 

76.1838 

461.864 

30 

94.2480 

706.860 

24% 

76.5765 

466.638 

30% 

94.6407 

712.763 

24% 

76.9692 

471.436 

30% 

95.0334 

718.690 

24% 

77.3619 

476.259 

30% 

95.4261 

724.642 

24% 

77.7546 

481.107 

30% 

95.8188 

730.618 

24% 

78.1473 

485.979 

30% 

96.2115 

736.619 

25 

78.5400 

490.875 

30% 

96.6042 

742.645 

25% 

78.9327 

495.796 

30% 

96.9969 

748.695 

25% 

79.3254 

500.742 

31 

97.3896 

754.769 

25% 

79.7181 

505.712 

31% 

97.7823 

760.869 

25% 

80.1108 

510.706 

31% 

98.1750 

766.992 

25*| 

80.5035 

515.726 

31% 

98.5677 

773.140 

25% 

80.8962 

520.769 

31  y 

98.9604 

779.313 

25% 

81.2889 

525.838 

31^1 

99.3531 

785.510 

26 

81.6816 

530.930 

31% 

99.7458 

791.732 

26% 

82.0743 

536.048 

31% 

100.1385 

797.979 

26% 

82.4670 

541.190 

32 

100.5312 

804.250 

26% 

82.8597 

546.356 

32% 

100.9239 

810.545 

26% 

83.2524 

551.547 

32% 

101.3166 

816.865 

26% 

83.6451 

556.763 

32% 

101.7093  ' 

823.210 

26% 

84.0378 

562.003 

32% 

102.1020 

829.579 

26% 

84.4305 

567.267 

32% 

102.4947 

835.972 

27 

84.8232 

572.557 

32% 

102.8874 

842.391 

TABLE  OF  CIRCLES. 


85 


Table — ( Continued). 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

32% 

103.280 

848.833 

38% 

121.344 

1,171.731 

33 

103.673 

855.301 

38% 

121.737 

1,179.327 

33  % 

104.065 

861.792 

38% 

122.130 

1,186.948 

33% 

104.458 

868.309 

39 

122.522 

1,194.593 

33% 

104.851 

874.850 

39% 

122.915 

1,202.263 

33  % 

105.244 

881.415 

39% 

123.308 

1,209.958 

33% 

105.636 

888.005 

39% 

123.700 

1,217.677 

33% 

106.029 

894.620 

39% 

124.093 

1,225.420 

33% 

106.422 

901.259 

39% 

124.486 

1,233.188 

34 

106.814 

907.922 

39% 

124.879 

1,240.981 

34% 

107.207 

914.611 

39% 

125.271 

1,248.798 

34lJ 

107.600 

921.323 

40 

125.664 

1,256.640 

34% 

107.992 

928.061 

40% 

126.057 

1,264.510 

34% 

108.385 

934.822 

40l| 

126.449 

1,272.400 

34% 

108.778 

941.609 

40% 

126.842 

1,280.310 

34% 

109.171 

948.420 

40% 

127.235 

1,288.250 

34% 

109.563 

955.255 

40% 

127.627 

1,296.220 

35 

109.956 

962.115 

40% 

128.020 

1,304.210 

35% 

110.349 

969.000 

40% 

128.413 

1,312.220 

35% 

110.741 

975.909 

41 

128.806 

1,320.260 

35^1 

111.134 

982.842 

41% 

129.198 

1,328.320 

35% 

111.527 

989.800 

41% 

129.591 

1,336.410 

35% 

111.919 

996.783 

41% 

129.984 

1,344.520 

35% 

112.312 

1,003.790 

41% 

130.376 

1,352.660 

112.705 

1,010.822 

41% 

130.769 

1,360.820 

36 

113.098 

1,017.878 

41% 

131.162 

1,369.000 

36% 

113.490 

1,024.960 

41% 

131.554 

1,377.210 

36% 

113.883 

1,032.065 

42 

131.947 

1,385.450 

36% 

114.276 

1,039.195 

42% 

132.340 

1,393.700 

36% 

114.668 

1,046.349 

42% 

132.733 

1,401.990 

36% 

115.061 

1,053.528 

423% 

133.125 

1,410.300 

36% 

115.454 

1,060.732 

42% 

133.518 

1,418.630 

36% 

115.846 

1,067.960 

42% 

133.911 

1,426.990 

37 

116.239 

1,075.213 

42% 

134.303 

1,435.370 

37% 

116.632 

1,082.490 

42% 

134.696 

1,443.770 

37% 

117.025 

1,089.792 

43 

135.089 

1,452.200 

37% 

117.417 

1,097.118 

43% 

135.481 

1,460.660 

37% 

117.810 

1,104.469 

43% 

135.874 

1,469.140 

37% 

118.203 

1,111.844 

43% 

136.267 

1,477.640 

37% 

118.595 

1,119.244 

43% 

136.660 

1,486.170 

37% 

118.988 

1,126.669 

43% 

137.052 

1,494.730 

38 

119.381 

1,134.118 

43% 

137.445 

1,503.300 

38% 

119.773 

1,141.591 

43% 

137.838 

1,511.910 

38% 

120.166 

1,149.089 

44 

138.230 

1,520.530 

38% 

120.559 

1,156.612 

44% 

138623 

1,529.190 

38% 

120.952 

1,164.159 

44% 

139.016 

1,537.860 

86 


USEFUL  TABLES. 


Table — ( Continued). 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

8 

139.408 

1,546.56 

50% 

157.473 

1,973.33 

139.801 

1,555.29 

50% 

157.865 

1,983.18 

140.194 

1,564.04 

50% 

158.258 

1,993.06 

44 % 
44% 

140.587 

1,572.81 

50% 

158.651 

2,002.97 

140.979 

1,581.61 

50% 

159.043 

2,012.89 

45 

141.372 

1,590.43 

50% 

159.436 

2,022.85 

45% 

141.765 

1,599.28 

50% 

159.829 

2,032.82 

45% 

142.157 

1,608.16 

51 

160.222 

2,042.83 

45% 

142.550 

1,617.05 

51% 

160.614 

2,052.85 

4^1 

45% 

142.943 

143.335 

1,625.97 

1,634.92 

if 

51% 

51% 

161.007 

161.400 

2,062.90 

2,072.98 

143.728 

1,643.89 

161.792 

2,083.08 

144.121 

1,652.89 

162.185 

2,093.20 

46 

144  514 

1,661.91 

51% 

162.578 

2,103.35 

46% 

144.906 

1,670.95 

51% 

162.970 

2,113.52 

4fil| 

145.299 

1,680.02 

52 

163.363 

2,123.72 

46% 

145.692 

1,689.11 

52% 

52% 

163.756 

2,133.94 

46% 

146.084 

1,698.23 

164.149 

2,144.19 

46% 

146.477 

1,707.37 

52% 

164.541 

2,154.46 

46k 

146.870 

1,716.54 

52% 

164.934 

2,164.76 

46% 

147.262 

1,725.73 

52% 

165.327 

2,175.08 

47 

147.655 

1,734.95 

52% 

165.719 

2,185.42 

47  % 

148.048 

1,744.19 

52% 

166.112 

2,195.79 

SB 

148.441 

1,753.45 

53 

166.505 

2,206.19 

148.833 

1,762.74 

53% 

166.897 

2,216.61 

47% 

149.226 

1,772.06 

II 

167.290 

2,227.05 

47k 

149.619 

1,781.40 

167.683 

2,237.52 

47^ 

150.011 

1,790.76 

53% 

168.076 

2,248.01 

47% 

150.404 

1,800.15 

53^ 

168.468 

2,258.53 

48 

150.797 

1,809.56 

53^1 

53j| 

168.861 

2,269.07 

48% 

151.189 

1,819.00 

169.254 

2,279.64 

48% 

151.582 

1,828.46 

54 

169.646 

2,290.23 

48^ 

151.975 

1,837.95 

54% 

170.039 

2,300.84 

152.368 

1,847.46 

54% 

170.432 

2,311.48 

48% 

152.760 

1,856.99 

54% 

170.824 

2,322.15 

48% 

153.153 

1,866.55 

54% 

171.217 

2,332.83 

48% 

153.546 

1,876.14 

54% 

171.610 

2,343.55 

49 

153.938 

1,885.75 

54% 

172.003 

2,354.29 

49% 

154.331 

1,895.38 

54% 

172.395 

2,365.05 

.SB 

154.724 

1,905.04 

55 

172.788 

2,375.83 

155.116 

1,914.72 

55% 

173.181 

2,386.65 

SB 

St 

155.509 

1,924.43 

55% 

173.573 

2,397.48 

155.902 

1,934.16 

55% 

173.966 

2,408.34 

156.295 

1,943.91 

55% 

174.359 

2,419.23 

156.687 

1,953.69 

55% 

174.751 

2,430.14 

50 

157.080 

1,963.50 

55% 

175.144 

2,441.07 

TABLE  OF  CIRCLES. 


87 


Table— ( Continued). 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

55% 

175.537 

2,452.03 

61% 

61% 

193.601 

2,982.67 

56 

175.930 

2,463.01 

193.994 

2,994.78 

176.322 

2,474.02 

61% 

194.386 

3,006.92 

5634 

176.715 

2,485.05 

62 

194.779 

3,019.08 

56% 

177.108 

2,496.11 

62% 

195.172 

3,031.26 

56% 

177.500 

2,507.19 

62% 

195.565 

3,043.47 

56% 

177.893 

2,518.30 

62% 

195.957 

3,055.71 

sak 

178.286 

2,529.43 

62% 

196.350 

3,067.97 

56% 

178.678 

2,540.58 

62% 

196.743 

3,080.25 

57 

179.071 

2,551.76 

62% 

197.135 

3,092.56 

57% 

179.464 

2,562.97 

62% 

197.528 

3,104.89 

57% 

179.857 

2,574.20 

63 

197.921 

3,117.25 

57$ 

180.249 

2,585.45 

63^ 

198.313 

3,129.64 

57k 

180.642 

2,596.73 

198.706 

3,142.04 

57% 

181.035 

2,608.03 

63^1 

199.099 

3,154.47 

57% 

181  427 

2,619.36 

63% 

199.492 

3,166.93 

57JI 

181.820 

2,630.71 

63% 

199.884 

3,179.41 

58 

182.213 

2,642.09 

63% 

200.277 

3,191.91 

58% 

182.605 

2,653.49 

63% 

200.670 

3,204.44 

58% 

182.998 

2,664.91 

64 

201.062 

3,217.00 

58|| 

183.391 

2,676.36 

64% 

201.455 

3,229.58 

183.784 

2,687.84 

64% 

201.848 

3,242.18 

58% 

184.176 

2,699.33 

64% 

202.240 

3,254.81 

58% 

184.569 

2,710.86 

64% 

202.633 

3,267.46 

58% 

184.962 

2,722.41 

64% 

203.026 

3,280.14 

59 

185.354 

2,733.98 

64% 

203.419 

3,292.84 

59% 

185.747 

2,745.57 

64% 

203.811 

3,305.56 

59% 

186.140 

2,757.20 

65 

204.204 

3,318.31 

59% 

186.532 

2,768.84 

65% 

204.597 

3,331.09 

59% 

186.925 

2,780.51 

65% 

204.989 

3,343.89 

59% 

187.318 

2,792.21 

65% 

205.382 

3,356.71 

59% 

187.711 

2,803.93 

65% 

205.775 

3,369.56 

59% 

188.103 

2,815.67 

65% 

206.167 

3,382.44 

60 

188.496 

2,827.44 

65% 

206.560  • 

3,395.33 

60% 

188.889 

2,839.23 

65% 

206.953 

3,408.26 

60% 

189.281 

2,851.05 

66 

207.346 

3,421.20 

60% 

189.674 

2,862.89 

66% 

207.738 

3,434.17 

60% 

190.067 

2,874.76 

66% 

208.131 

3,447.17 

60% 

190.459 

2,886.65 

66% 

208.524 

3,460.19 

60% 

190.852 

2,898.57 

66% 

208.916 

3,473.24 

60% 

191.245 

2,910.51 

66% 

209.309 

3,486.30 

61 

191.638 

2,922.47 

66% 

209.702 

3,499.40 

61% 

192.030 

2,934.46 

66% 

210.094 

3,512.52 

61% 

192.423 

2,946.48 

67 

210.487 

3,525.66 

61% 

192.816 

2,958.52 

67% 

210.880 

3,538.83 

61% 

193.208 

2,970.58 

67% 

211.273 

3,552.02 

88 


USEFUL  TABLES. 


Table — ( Continued). 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

67% 

211.665 

3,565.24 

73% 

229.729 

4,199.74 

67% 

212.058 

3,578.48 

73% 

230.122 

4,214.11 

67% 

212.451 

3,591.74 

73% 

230.515 

4,228.51 

67% 

212.843 

3,605.04 

73% 

230.908 

4,242.93 

67% 

213.236 

3,618.35 

73% 

231.300 

4,257.37 

68 

213.629 

3,631.69 

73% 

231.693 

4,271.84 

68% 

214.021 

3,645.05 

73% 

232.086 

4,286.33 

68% 

214.414 

3,658.44 

74 

232.478 

4,300.85 

68% 

214.807 

3,671.86 

74% 

232.871 

4,315.39 

68% 

215.200 

3,685.29 

74% 

233.264 

4,329.96 

68|| 

215.592 

3,698.76 

74% 

233.656 

4,344.55 

Ws 

215.985 

3,712.24 

74% 

234.049 

4,359.17 

216.378 

3,725.75 

74% 

234.442 

4,373.81 

69 

216.770 

3,739.29 

74% 

234.835 

4,388.47 

69Ys 

217.163 

3,752.85 

74% 

235.227 

4,403.16 

69% 

217.556 

3,766.43 

75 

235.620 

4,417.87 

69% 

217.948 

3,780.04 

75% 

236.013 

4,432.61 

69% 

218.341 

3,793.68 

75% 

236.405 

4,447.38 

69% 

218.734 

3,807.34 

75% 

236.798 

4,462.16 

69% 

219.127 

3,821.02 

75% 

237.191 

4,476.98 

69% 

219.519 

3,834.73 

75% 

237.583 

4,491.81 

70 

219.912 

3,848.46 

75% 

237.976 

4,506.67 

70% 

220.305 

3,862.22 

75% 

238.369 

4,521.56 

70^1 

220.697 

3,876.00 

76 

238.762 

4,536.47 

70% 

221.090 

3,889.80 

76% 

239.154 

4,551.41 

70% 

221.483 

3,903.63 

76% 

239.547 

4,566.36 

70% 

221.875 

3,917.49 

76% 

239.940 

4,581.35 

70% 

222.268 

3,931.37 

76% 

240.332 

4,596.36 

70% 

222.661 

3,945.27 

76% 

240.725 

4,611.39 

71 

223.054 

3,959.20 

76% 

241.118 

4,626.45 

71% 

223.446 

3,973.15 

76% 

241.510 

4,641.53 

71% 

223.839 

3,987.13 

77 

241.903 

4,656.64 

71% 

224.232 

4,001.13 

77% 

242.296 

4,671.77 

71% 

224.624 

4,015.16 

77% 

242.689 

4,686.92 

71% 

225.017 

4,029.21 

77% 

243.081 

4,702.10 

71% 

225.410 

4,043.29 

77% 

77% 

243.474 

4,717.31 

71% 

225.802 

4,057.39 

243.867 

4,732.54 

72 

226.195 

4,071.51 

77% 

77% 

244.259 

4,747.79 

72% 

226.588 

4,085.66 

244.652 

4,763.07 

72% 

226.981 

4,099.84 

78 

245.045 

4,778.37 

72% 

227.373 

4,114.04 

78% 

245.437 

4,793.70 

72% 

227.766 

4,128.26 

78% 

245.830 

4,809.05 

72% 

228.159 

4,142.51 

78% 

246.223 

4,824.43 

72% 

228.551 

4,156.78 

78% 

246.616 

4,839.83 

72% 

228.944 

4,171.08 

78% 

247.008 

4,855.26 

73 

229.337 

4,185.40 

78% 

247.401 

4,870.71 

TABLE  OF  CIRCLES. 


89 


Table— ( Continued). 


Diam. 

Circum. 

Area. 

Diam. 

Circum. 

Area. 

78k 

247.794 

4,886.18 

84V 

265.858 

5,624.56 

79 

248.186 

4,901.68 

84V 

266.251 

5,641.18 

79k 

248.579 

4,917.21 

84k 

266.643 

5,657.84 

79V 

248.972 

4,932.75 

85 

267.036 

5,674.51 

79k 

249.364 

4,948.33 

85k 

267.429 

5,691.22 

79V 

249.757 

4,963.92 

85V 

267.821 

5,707.94 

79% 

250.150 

4,979.55 

85k 

268.214 

5,724.69 

79% 

250.543 

4,995.19 

85  V 

268.607 

5,741.47 

79j| 

250.935 

5,010.86 

85k 

268.999 

5,758.27 

80 

251.328 

5,026.56 

85V 

269.392 

5,775.10 

80k 

251.721 

5,042.28 

85k 

269.785 

5,791.94 

80V 

252.113 

5,058.03 

86 

270.178 

5,808.82 

80k 

252.506 

5,073.79 

86k 

270.570 

5,825.72 

80V 

252.899 

5,089.59 

86V 

270.963 

',842.64 

80 % 

253.291 

5,105.41 

86k 

271.356 

5,859.59 

lo^l 

253.684 

5,121.25 

86k 

271.748 

5,876.56 

254.077 

5,137.12 

86k 

272.141 

5,893.55 

81 

254.470 

5,153.01 

86V 

272.534 

5,910.58 

81^ 

254.862 

5,168.93 

86j| 

272.926 

5,927.62 

255.255 

5,184.87 

87 

273.319 

5,944.69 

81k 

255.648 

5,200.83 

87k 

273.712 

5,961.79 

8 iv 

256.040 

5,216.82 

87k 

274.105 

5,978.91 

81k 

256.433 

5,232.84 

87k 

274.497 

5.996.05 

81V 

256.826 

5,248.88 

87V 

274.890 

6,013.22 

81k 

257.218 

5,264.94 

87^| 

275.283 

6,030.41 

82 

257.611 

5,281.03 

87V 

275.675 

6,047.63 

82k 

258.004 

5,297.14 

87j| 

276.068 

6,064.87 

82V 

258.397 

5,313.28 

88 

276.461 

6,082.14 

82k 

82V 

258.789 

5,329.44 

88k 

276.853 

6,099.43 

259.182 

5,345.63 

88V 

277.246 

6,116.74 

82k 

259.575 

5,361.84 

88k 

277.629 

6,134.08 

82V 

259.967 

5,378.08 

88V 

278.032 

6,151.45 

82% 

260.360 

5,394.34 

88^ 

278.424 

6,168.84 

83 

250.753 

5,410.62 

88V 

278.817 

6,186.25 

83k 

261.145 

5,426.93 

88k 

279.210 

6,203.69 

83V 

261.538 

5,443.26 

89 

279.602 

6,221.15 

83k 

261.931 

5,459.62 

89k 

279.995 

6,238.64 

83V 

262.324 

5,476.01 

89V 

280.388 

6,256.15 

83 f8 

262.716 

5,492.41 

89k 

280.780 

6,273.69 

263.109 

5,508.84 

89k 

281.173 

6,291.25 

83k 

263.502 

5,525.30 

89k 

281.566 

6,308.84 

84 

263.894 

5,541.78 

89V 

281.959 

6,326.45 

84V 

264.287 

5,558.29 

89k 

282.351 

6,344.08 

264.680 

5,574.82 

90 

282.744 

6,361.74 

84/ls 

265.072 

5,591.37 

90k 

283.137 

6,379.42 

84k 

265.465 

5,607.95 

90k 

283.529 

6,397.13 

90 


USEFUL  TABLES. 


T a b le — ( Continued ) . 


Diam. 

Circum, 

Area. 

Diam. 

Circum. 

Area. 

90% 

283.922 

6,414.86 

95% 

299.237 

7,125.59 

90% 

284.315 

6,432.62 

95% 

299.630 

7,144.31 

90% 

284.707 

6,450.40 

95% 

300.023 

7,163.04 

90% 

285.100 

6,468.21 

95% 

300.415 

7,181.81 

285.493 

6,486.04 

95% 

300.808 

7,200.60 

91 

285.886 

6,503.90 

95% 

301.201 

7,219.41 

91% 

286.278  ” 

6,521.78 

96 

301.594 

7,238.25 

91% 

286.671 

6,539.68 

96% 

301.986 

7,257.11 

91ft 

287.064 

6,557.61 

96% 

302.379 

7,275.99 

287.456 

6,575.56 

96% 

302.772 

7,294.91 

91% 

287.849 

6,593.54 

96% 

303.164 

7,313.84 

91% 

288.242 

6,611.55 

96% 

303.557 

7,332.80 

91% 

288.634 

6,629.57 

96% 

303.950 

7,351.79 

92 

289.027 

6,647.63 

96% 

304.342 

7,370.79 

92% 

289.420 

6,665.70 

97 

304.735 

7,389.83 

92% 

289.813 

6,683.80 

97% 

305.128 

7,408.89 

92% 

290.205 

6,701.93 

97% 

305.521 

7,427.97 

92% 

290.598 

6,720.08 

97% 

305.913 

7,447.08 

92% 

290.991 

6,738.25 

97% 

306.306 

7,466.21 

92% 

291.383 

6,756.45 

97% 

306.699 

7,485.37 

92% 

291.776 

6,774.68 

97% 

307.091 

7,504.55 

93 

292.169 

6,792.92 

97% 

307.484 

7,523.75 

93% 

292.562 

6,811.20 

98 

307.877 

7,542.98 

93% 

292.954 

6,829.49 

98% 

308.270 

7,562.24 

93% 

293.347 

6,847.82 

98% 

308.662 

7,581.52 

293.740 

6,866.16 

98|| 

309.055 

7,600.82 

93% 

294.132 

6,884.53 

309.448 

7,620.15 

93% 

294.525 

6,902.93 

98% 

309.840 

7,639.50 

93% 

294.918 

6,921.35 

98% 

310.233 

7,658.88 

94 

295.310 

6,939.79 

98% 

310.626 

7,678.28 

94% 

295.703 

6,958.26 

99 

311,018 

7,697.71 

94% 

296.096 

6,976.76 

99% 

311.411 

7,717.16 

94% 

296.488 

6,995.28 

99% 

311.804 

7,736.63 

94% 

296.881 

7,013.82 

99% 

312.196 

7,756.13 

94% 

297.274 

7,032.39 

99% 

312.589 

7,775.66 

94% 

297.667 

7,050.98 

99% 

312.982 

7,795.21 

94% 

298.059 

7,069.59 

99% 

313.375 

7,814.78 

95 

298.452 

7,088.24 

99% 

313.767 

7,834.38 

95% 

298.845 

7,106.90 

100 

314.160 

7,854.00 

The  preceding  table  may  be  used  to  determine  the 
diameter  when  the  circumference  or  area  is  known.  Thus, 
the  diameter  of  a circle  having  an  area  of  7,200  sq.  in.  is, 
approximately,  95$  in. 


DECIMAL  EQUIVALENTS. 


91 


DECIMAL  EQUIVALENTS  OF  PARTS  OF  ONE  INCH. 


1-64 

.015625 

17-64 

.265625 

1-32 

.031250 

9-32 

.281250 

3-64 

.046875 

19-64 

.296875 

1-16 

.062500 

5-16 

.312500 

5-04 

.078125 

21-64 

.328125 

3-32 

.093750 

11-32 

.343750 

7-64 

.109375 

23-64 

.359375 

1-8 

.125000 

3-8 

.375000 

9-64 

.140625 

25-64 

.390625 

5-32 

.156250 

13-32 

.406250 

11-64 

.171875 

27-64 

.421875 

3-16 

.187500 

7-16 

.437500 

13-64 

.203125 

29-64 

.453125 

7-32 

.218750 

15-32 

.468750 

15-64 

.234375 

31-64 

.484375 

1-4 

.250000 

1-2 

.500000 

33-64 

.515625 

49-64 

.765625 

17-32 

.531250 

25-32 

.781250 

35-64 

.546875 

51-64 

.796875 

9-16 

.562500 

13-16 

.812500 

37-64 

.578125 

53-64 

.828125 

19-32 

.593750 

27-32 

.843750 

39-64 

.609375 

55-64 

.859375 

5-8 

.625000 

7-8 

.875000 

41-64 

.640625 

57-64 

.890625 

21-32 

.656250 

29-32 

.906250 

43-64 

.671875 

59-64 

.921875 

11-16 

.687500 

15-16 

.937500 

45-64 

.703125 

61-64 

.953125 

23-32 

.718750 

31-32 

.968750 

47-64 

.734375 

63-64 

.984375 

3-4 

.750000 

1 

1 

DECIMALS  OF  A FOOT  FOR  EACH  1-32  OF  AN  INCH. 


Inch. 

0" 

1" 

2" 

3" 

4" 

5" 

0 

0 

.0833 

.1667 

.2500 

.3333 

.4167 

.0026 

.0859 

.1693 

.2526 

.3359 

.4193 

A 

.0052 

.0885 

.1719 

.2552 

.3385 

.4219 

V? 

.0078 

.0911 

.1745 

.2578 

.3411 

.4245 

V& 

.0104 

.0937 

.1771 

.2604 

.3437 

.4271 

& 

.0130 

.0964 

.1797 

.2630 

.3464 

.4297 

A 

.0156 

.0990 

.1823 

.2656 

.3490 

.4323 

& 

.0182 

.1016 

.1849 

.2682 

.3516 

.4349 

M 

.0208 

.1612 

.1875 

.2708 

.3542 

.4375 

.0234 

.1068 

.1901 

.2734 

.3568 

.4401 

.0260 

.1094 

.1927 

.2760 

.3594 

.4427 

II 

.0286 

.1120 

.1953 

.2786 

.3620 

.4453 

h 

.0312 

.1146 

.1979 

.2812 

.3646 

.4479 

.0339 

.1172 

.2005 

.2839 

.3672 

.4505 

TV 

.0365 

.1198 

.2031 

.2865 

.3698 

.4531 

d 

.0391 

.1224 

.2057 

.2891 

.3724 

.4557 

% 

.0417 

.1250 

.2083 

.2917 

.3750 

.4583 

& 

.0443 

.1276 

.2109 

.2943 

.3776 

.4609 

TJS 

.0469 

.1302 

.2135 

.2969 

.3802 

.4635 

If 

.0495 

.1328 

.2161 

.2995 

.3828 

.4661 

% 

.0521 

.1354 

.2188 

.3021 

.3854 

.4688 

u 

.0547 

.1380 

.2214 

.3047 

.3880 

.4714 

H 

.0573 

.1406 

.2240 

.3073 

.3906 

.4740 

it 

.0599 

.1432 

.2266 

.3099 

.3932 

.4766 

92 


USEFUL  TABLES. 


Table— ( Continued). 


Inch. 

0" 

1" 

2" 

3" 

4" 

5". 

% 

.0625 

.1458 

.2292 

.3125 

.3958 

.4792 

if 

.0651 

.1484 

.2318 

.3151 

.3984 

.4818 

H 

.0677 

.1510 

.2344 

.3177 

.4010 

.4844 

27 

.0703 

.1536 

.2370 

.3203 

.4036 

.4870 

/I 

.0729 

.1562 

.2396 

.3229 

.4062 

.4896 

§7 

.0755 

.1589 

.2422 

.3255 

.4089 

.4922 

|| 

.0781 

.1615 

.2448 

.3281 

.4115 

.4948 

§7 

.0807 

.1641 

.2474 

.3307 

.4141 

.4974 

DECIMALS  OF  A FOOT  FOR  EACH  1-32  OF  AN  INCH, 


Inch. 

6" 

7" 

8" 

9" 

10" 

11" 

0 

.5000 

.5833 

.6667 

.7500 

.8333 

.9167 

A 

.5026 

.5859 

.6693 

.7526 

.8359 

.9193 

.5052 

.5885 

.6719 

.7552 

.8385 

.9219  ' 

ft 

.5078 

.5911 

.6745 

.7578 

.8411 

.9245 

.5104 

.5937 

.6771 

.7604 

.8437 

.9271 

.5130 

.5964 

.6797 

.7630 

.8464 

.9297 

’re 

.5156 

.5990 

.6823 

.7656 

.8490 

.9323 

.5182 

.6016 

.6849 

.7682 

.8516 

.9349 

u 

.5208 

.6042 

.6875 

.7708 

.8542 

.9375 

& 

.5234 

.6068 

.6901 

.7734 

.8568 

.9401 

V 

re 

.5260 

.6094 

.6927 

.7760 

.8594 

.9427 

1 1 
32 

.5286 

.6120 

.6953 

.7786 

.8620 

.9453 

% 

.5312 

.6146 

.6979 

.7812 

.8646 

.9479 

M 

re 

.5339 

.6172 

.7005 

.7839 

.8672 

.9505 

.5365 

.6198 

.7031 

.7865 

.8698 

.9531 

1 6 
32 

.5391 

.6224 

.7057 

.7891 

.8724 

.9557 

.5417 

.6250 

.7083 

.7917 

.8750 

.9583 

§7 

.5443 

.6276 

.7109 

.7943 

.8776 

.9609 

re 

.5469 

.6302 

.7135 

.7969 

.8802 

.9635 

.5495 

.6328 

.7161 

.7995 

.8828 

.9661 

.5521 

.6354 

.7188 

.8021 

.8854 

.9688 

14 

.5547 

.6380 

.7214 

.8047 

.8880 

.9714 

re 

.5573 

.6406 

.7240 

.8073 

.8906 

.9740 

?§ 

.5599 

.6432 

.7266 

.8099 

.8932 

.9766 

% 

.5625 

.6458 

.7292 

.8125 

.8958 

.9792 

§1 

.5651 

.6484 

.7318 

.8151 

.8984 

.9818 

.5677 

.6510 

.7344 

.8177 

.9010 

.9844 

S 

.5703 

.6536 

.7370 

.8203 

.9036 

.9870 

.5729 

.6562 

.7396 

.8229 

.9062 

.9896 

.5755 

.6589 

.7422 

.8255 

.9089 

.9922 

.5781 

.6615 

.7448 

.8281 

.9115 

.9948 

31 

52 

.5807 

.6641 

.7474 

.8307 

.9141 

.9974 

FORMULAS. 


93 


FORMULAS. 


= {+[-:(V/X/h-):-]|  = 

The  term  formula , as  used  in  mathematics  and  in  techni- 
cal books,  may  be  defined  as  a rule  in  which  symbols  are  used 
instead  of  words;  in  fact,  a formula  may  be  regarded  as  a 
shorthand  method  of  expressing  a rule. 

Most  people  having  no  knowledge  of  algebra  regard  for- 
mulas with  distrust;  they  think  that  a person  must  be  a good 
algebraic  scholar  in  order  to  be  able  to  use  formulas.  This 
idea,  however,  is  erroneous.  As  a rule,  no  knowledge  of 
any  branch  of  mathematics  except  arithmetic  is  required  to 
enable  one  to  use  a formula.  Any  formula  can  be  expressed 
in  words,  and  when  so  expressed  it  becomes  a rule. 

Formulas  are  much  more  convenient  than  rules;  they  show 
at  a glance  all  the  operations  that  are  to  be  performed;  they 
do  not  require  to  be  read  three  or  four  times,  as  is  the  case 
with  most  rules,  to  enable  one  to  understand  their  meaning; 
they  take  up  much  less  space,  both  in  the'  printed  book  and 
in  one’s  note  book,  than  rules;  in  short,  whenever  a rule  can 
be  expressed  as  a formula,  the  formula  is  to  be  preferred.  In 
the  following  pages  we  purpose  to  show  the  reader  how  to 
use  such  formulas  as  he  is  likely  to  encounter  in  “pocket- 
books,”  or  other  works  of  like  nature. 

The  signs  used  in  formulas  are  the  ordinary  signs  indica- 
tive of  operations  and  the  signs  of  aggregation.  All  these 
signs  are  used  in  arithmetic,  but,  to  refresh  the  reader’s 
memory,  we  will  explain  their  nature  and  uses  before  pro- 
ceeding further. 

The  signs  indicative  of  operations  are  six  in  number,  viz.: 

+ » > X,  -T-,  | , ~V  • 

The  sign  ( + ) indicates  addition,  and  is  called  plus;  when 
placed  between  two  quantities,  it  indicates  that  the  two 
quantities  are  to  be  added.  Thus,  in  the  expression  25  + 17, 
the  sign  ( + ) shows  that  17  is  to  be  added  to  25. 

The'  sign  (— ) indicates  subtraction,  and  is  called  minus; 
when  placed  between  two  quantities,  it  indicates  that  the 


94 


FORMULAS. 


quantity  on  the  right  is  to  be  subtracted  from  that  on  the 
left.  Thus,  in  the  expression  25  — 17,  the  sign  (— ) shows  that 
17  is  to  be  subtracted  from  25. 

The  sign  (x)  indicates  multiplication,  and  is  read  times , or 
multiplied  by;  when  placed  between  two  quantities,  it  indi- 
cates that  the  quantity  on  the  left  is  to  be  multiplied  by  that 
on  the  right.  Thus,  in  the  expression  25  X 17,  the  sign  (X) 
shows  that  25  is  to  be  multiplied  by  17. 

The  sign  (-^)  indicates  division,  and  is  read  divided  by; 
when  placed  between  two  quantities,  it  indicates  that  the 
quantity  on  the  left  is  to  be  divided  by  that  on  the  right. 
Thus,  in  the  expression  25  -f- 17,  the  sign  (^)  shows  that  25  is 
to  be  divided  by  17. 

Division  is  also  indicated  by  placing  a straight  line 
between  the  two  quantities.  Thus,  25  1 17,  25/17,  and  ff  all 
indicate  that  25  is  to  be  divided  by  17.  When  both  quantities 
are  placed  on  the  same  horizontal  line,  the  straight  line 
indicates  that  the  quantity  on  the  left  is  to  be  divided  by 
that  on  the  right.  When  one  quantity  is  below  the  other,  the 
straight  line  between  indicates  that  the  quantity  above  the 
line  is  to  be  divided  by  the  one  below  it. 

The  sign  (/)  indicates  that  some  root  of  the  quantity  to 
the  right  is  to  be  taken;  it  is  called  the  radical  sign.  To 
indicate  what  root  is  to  be  taken,  a small  figure,  called  the 
index , is  placed  within  the  sign,  this  being  always  omitted 
when  the  square  root  is  to  be  indicated.  Thus,  y/  25  indicates 
that  the  square  root  of  25  is  to  be  taken;  ^ 25  indicates  that 
the  cube  root  of  25  is  to  be  taken,  etc. 

Note.— As  the  term  “quantity”  is  a very  convenient  one 
to  use,  we  will  define  it.  In  mathematics  the  word  quantity 
is  applied  to  anything  that  it  is  desired  to  subject  to  the 
ordinary  operations  of  addition,  subtraction,  multiplication, 
etc.,  when  we  do  not  wish  to  be  more  specific  and  state 
exactly  what  the  thing  is.  Thus,  we  can  say  “two  or  more 
numbers,”  or  “ two  or  more  quantities.”  The  word  quantity 
is  more  general  in  its  meaning  than  the  word  number. 

The  signs  of  aggregation  are  four  in  number,  viz.: , 

(),  [],and  | respectively  called  the  vinculum , the  paren- 
thesis, the  brackets , and  the  brace;  they  are  used  when  it  is 
desired  to  indicate  that  all  the  quantities  included  by  them 


FORMULAS. 


95 


are  to  be  subjected  to  the  same  operation.  Thus,  if  we  desire 
to  indicate  that  the  sum  of  5 and  8 is  to  be  multiplied  by  7, 
and  we  do  not  wish  to  actually  add  5 and  8 before  indicating 
the  multiplication,  we  may  employ  any  one  of  the  four  signs 
of  aggregation  as  here  shown : 5 + 8 X 7,  ( 5 + 8 ) X 7,  [5  + 8]  X 7, 
\ 5 + 8 1 X 7.  The  vinculum  is  placed  above  the  quantities 
which  are  to  be  treated  as  one  quantity  and  subjected  to  the 
same  operations. 

While  any  one  of  the  four  signs  may  be  used  as  shown 
above,  custom  has  restricted  their  use  somewhat.  The  vincu- 
lum is  rarely  used  except  in  connection  with  the  radical  sign. 
Thus,  instead  of  writing  1^(5  + 8),  [5  + 8],  or  1^5 + 8| 

for  the  cube  root  of  5 plus  8,  all  of  which  would  be  correct, 
the  vinculum  is  nearly  always  used,  ^5  + 8. 

In  cases  where  but  one  sign  of  aggregation  is  needed 
(except,  of  course,  when  a root  is  to  be  indicated),  the 
parenthesis  is  always  used.  Hence,  (5  + 8)  X 7 would  be  the 
usual  way  of  expressing  the  product  of  5 plus  8 and  7. 

If  two  signs  of  aggregation  are  needed,  the  brackets  and 
parenthesis  are  used,  so  as  to  avoid  having  a parenthesis 
within  a parenthesis,  the  brackets  being  placed  outside.  For 
example,  [(20  — 5)  -i-  3]  X 9 means  that  the  difference  between 
20  and  5 is  to  be  divided  by  3,  and  this  result  multiplied  by  9. 

If  three  signs  of  aggregation  are  required,  the  brace, 
brackets,  and  parenthesis  are  used,  the  brace  being  placed 
outside,  the  brackets  next,  and  the  parenthesis  inside.  For 
example,  \ [(20  — 5)  -4-3]  X9  — 21  £ -4-8  means  that  the  quo- 
tient obtained  by  dividing  the  difference  between  20  and  5 
by  3 is  to  be  multiplied  by  9;  and  that  21  is  to  be  subtracted 
from  the  product  thus  obtained,  and  the  result  divided  by  8. 

Should  it  be  necessary  to  use  all  four  signs  of  aggrega- 
tion, the  brace  would  be  put  outside,  the  brackets  next, 
the  parenthesis  next,  and  the  vinculum  inside.  For  example, 
J [(20  — 5 3)  X 9 — 21]  -4-  8 £ X 12.  The  reason  for  using  the 

brace  in  this  last  instance  will  be  explained,  as  it  is  not 
generally  understood. 

When  several  quantities  are  connected  by  the  various 
signs  indicating  addition,  subtraction,  multiplication,  and 
division,  the  operation  indicated  by  the  sign  of  multiplication 


96 


FORMULAS. 


must  always  be  performed  first.  Thus,  2 + 3X4  equals 
14,  3 being  multiplied  by  4 before  adding  to  2.  Similarly, 
10  -f-  2 X 5 equals  1,  since  2X5  equals  10,  and  10  -i- 10  equals  1. 
Hence,  in  the  above  case,  if  the  brace  were  omitted,  the 
Tesult  would  be  + whereas,  by  inserting  the  brace,  the 
result  is  36. 

Following  the  sign  of  multiplication  comes  the  sign  of 
division  in  its  order  of  importance.  For  example,  5 — 9 3 

equals  2,  9 being  divided  by  3 before  subtracting  from  5.  The 
signs  of  addition  and  subtraction  are  of  equal  value;  that  is, 
if  several  quantities  are  connected  by  plus  and  minus  signs, 
the  indicated  operations  may  be  performed  in  the  order  in 
which  the  quantities  are  placed. 

There  is  one  other  sign  used,  which  is  neither  a sign  of 
aggregation  nor  a sign  indicative  of  an  operation  to  be  per- 
formed; it  is  (=),  and  is  called  the  sign  of  equality;  it  means 
that  all  on  one  side  of  it  is  exactly  equal  to  all  on  the  other 
side.  For  example,  2 = 2,  5 — 3 = 2,  5 X (14  — 9)  =25. 

Having  described  the  signs  used  in  formulas,  the  formulas 
themselves  will  now  be  explained.  First  consider  the  well- 
known  rule  for  finding  the  horsepower  of  a steam  engine, 
which  may  be  stated  as  follows : 

Divide  the  continued  product  of  the  mean  effective  pressure  in 
pounds  per  square  inch , the  length  of  the  stroke  in  feet , the  area 
of  the  piston  in  square  inches , and  the  number  of  strokes  per 
minute  by  83,000 ; the  result  will  be  the  horsepower. 

This  is  a very  simple  rule,  and  very  little,  if  anything,  will 
be  saved  by  expressing  it  as  a formula,  so  far  as  clearness  is 
concerned.  The  formula,  however,  will  occupy  a great  deal 
less  space,  as  we  shall  show. 

An  examination  of  the  rule  will  show  that  four  quantities ' 
(viz.,  the  mean  effective  pressure,  the  length  of  the  stroke, 
the  area  of  the  piston,  and  the  number  of  strokes)  are  multi- 
plied together,  and  the  result  is  divided  by  33,000.  Hence,  the 
rule  might  be  expressed  as  follows : 

mean  effective  pressure  v „ stroke 
(in  pounds  per  square  inch)  (in  feet) 
area  of  piston  number  of  strokes  ^ 

(in  square  inches)  (per  minute)  * ’ 


FORMULAS. 


07 


This  expression  could  be  shortened  by  representing  each 
quantity  by  a single  letter,  thus:  representing  horsepower 
by  the  letter  the  mean  effective  pressure  in  pounds  per 
square  inch  by  “P,”  the  length  of  the  stroke  in  feet  by  “X,” 
the  area  of  the  piston  in  square  inches  by  “A,”  the  number  of 
strokes  per  minute  by  “iV,”  and  substituting  these  letters  for 
the  quantities  that  they  represent,  the  above  expression 
would  reduce  to 

PXLXAXN 

33,000 

a much  simpler  and  shorter  expression.  This  last  expression 
is  called  a formula. 

The  formula  just  given  shows,  as  we  stated  in  the  begin- 
ning, that  a formula  is  really  a shorthand  method  of  express^ 
ing  a rule.  It  is  customary,  however,  to  omit  the  sign  of 
multiplication  between  two  or  more  quantities  when  they 
are  to  be  multiplied  together,  or  between  a number  and  a 
letter  representing  a quantity,  it  being  always  understood 
that  when  two  letters  are  adjacent  with  no  sign  between 
them,  the  quantities  represented  by  these  letters  are  to  be 
multiplied.  Bearing  this  fact  in  mind,  the  formula  just 
given  can  be  further  simplified  to 
PLAN 
11  ~ 33,000  * 

The  sign  of  multiplication,  evidently,  cannot  be  omitted 
between  two  or  more  numbers,  as  it  wrould  then  be  impossible 
to  distinguish  the  numbers.  A near  approach  to  this,  how- 
ever, may  be  attained  by  placing  a dot  between  the  numbers 
that  are  to  be  multiplied  together,  and  this  is  frequently  done 
in  works  on  mathematics  whep  it  is  desired  to  economize 
space.  . In  such  ca  ses  it  is  usual  to  put  the  dot  higher  than 
the  position  occupi  ^d  by  the  decimal  point.  Thus,  2-3  means 
the  same  as  2 X 3;  o42-749-l,006  indicates  that  the  numbers 
542,  749,  and  1,006  are  to  be  multiplied  together. 

It  is  also  customary  to  omit  the  sign  of  multiplication  in 
expressions  similar  to  the  following:  aX  l /b  + c,  3X  (b  + c), 
(b  + c)  X a,  etc.,  writing  them  a\/ b + c,  3(b  + c),  (b  + c)a,  etc. 
The  sign  is  not  omitted  when  several  quantities  are  included 
by  a vinculum,  and  it  is  desired  to  indicate  that  the  quantities 


93 


FORMULAS. 


so  included  are  to  be  multiplied  by  another  quantity. 
For  example,  3XHc,  6 + cX«,  4/b  + c X a,  etc.,  are 
always  written  as  here  printed. 

Before  proceeding  further,  we  will  explain  one  other 
device  that  is  used  by  formula  makers,  and  which  is  apt  to 
puzzle  one  who  encounters  it  for  the  first  time.  It  is  the 
use  of  what  mathematicians  call  primes  and  subs.,  and  what 
printers  call  superior  and  inferior  characters.  As  a rule, 
formula  makers  designate  quantities  by  the  initial  letters  of 
the  names  of  the  quantities.  For  example,  they  represent 
volume  by  v,  pressure  by  p,  height  by  h,  etc.  This  practice 
is  to  be  commended,  as  the  letter  itself  serves  in  many  cases 
to  identify  the  quantity  that  it  represents.  Some  authors 
carry  the  practice  a little  further  and  represent  all  quantities 
of  the  same  nature  by  the  same  letter  throughout  the  book, 
always  having  the  same  letter  represent  the  same  thing. 
Now,  this  practice  necessitates  the  use  of  the  primes  and 
subs,  above  mentioned  when  two  quantities  have  the  same 
name,  but  represent  different  things.  Thus,  consider  the 
word  pressure  as  applied  to  steam  at  different  stages  between 
the  boiler  and  the  condenser.  First,  there  is  absolute  pres- 
sure, which  is  equal  to  the  gauge  pressure  in  pounds  per 
square  inch  plus  the  pressure  indicated  by  the  barometer 
reading  (usually  assumed  in  practice  to  be  14.7  pounds  per 
square  inch,  when  a barometer  is  not  at  hand) . If  this  be 
represented  by  p,  how  shall  we  represent  the  gauge  pressure? 
Since  the  absolute  pressure  is  always  greater  than  the  gauge 
pressure,  suppose  we  decide  to  represent  it  by  a capital 
letter,  and  the  gauge  pressure  by  a small  (lower-case)  letter. 
Doing  so,  P represents  absolute  pressure,  and  p gauge  pres- 
sure. Further,  there  is  usually  a “drop”  in  pressure 
between  the  boiler  and  the  engine,  so  that  the  initial  pres- 
sure, or  pressure  at  the  beginning  of  the  stroke,  is  less  than 
the  pressure  at  the  boiler.  How  shall  we  represent  the 
initial  pressure?  We  may  do  this  in  one  of  three  ways,  and 
still  retain  the  letter  p or  P to  represent  the  word  pressure: 
First,  by  the  use  of  the  prime  mark;  thus,  p ' or  P'  (read  p 
prime  and  p major  prime)  may  be  considered  to  represent  the 
initial  gauge  pressure  or  the  initial  absolute  pressure. 


FORMULAS. 


99 


Second,  by  the  use  of  sub.  figures;  thus,  p\  or  Pi  (readp  sub. 
one  andp  major  sub.  one).  Third,  by  the  use  of  sub.  letters: 
thus,  pi  or  Pi  (read  p sub.  i and  P major  sub.  i).  Likewise,  p'1 
(read  p second),  p2 , or  pr  might  be  used  to  represent  the  gauge 
pressure  at  release,  etc.  Sub.  letters  have  the  advantage  of 
still  further  identifying  the  quantity  represented;  in  many 
instances,  however,  it  is  not  convenient  to  use  them,  in  which 
case  primes  and  subs,  are  used  instead.  The  prime  notation 
may  be  continued  as  follows:  p'",piy,py,  etc.;  it  is  inadvisable 
to  use  superior  figures,  for  example,  p1,  p2,  p3,  pa,  etc.,  as  they 
are  liable  to  be  mistaken  for  exponents. 

The  main  thing  to  be  remembered  by  the  reader  is  that 
when  a formula  is  given  in  which  the  same  letters  occur  several 
times,  all  like  letters  having  the  same  primes  or  subs,  represent  the 
same  quantities,  while  those  that  differ  in  any  respect  represent 
different  quantities.  Thus,  in  the  formula 

t __  W i Si  ti  + W2  So  h + w3  s3 13 
~ wx  Si  + Wo  s2  -}  i v3s3  * 

Wi,  w2,  and  w3  represent  the  weights  of  three  different  bodies; 
Si,s2,  and  s3  their  specific  heats;  and  ti,i*>,  and  t3  their  tem- 
peratures; while  t represents  the  final  temperature,  after  the 
bodies  have  been  mixed  together. 

It  is  very  easy  to  apply  the  above  formula  when  the 
values  of  the  quantities  represented  by  the  different  letters 
are  known.  All  that  is  required  is  to  substitute  the  numeri- 
cal values  of  the  letters,  and  then  perform  the  indicated 
operations.  Thus,  suppose  that  the  values  of  Si,  and  t\ 
are,  respectively,  2 pounds,  .0951,  and  80°;  of  w2,  s2,  and  for 
7.8  pounds,  1,  and  80°,  and  of  ws,  s3,  and  t3,  3£  pounds,  .1138, 
and  780°;  then,  the  final  temperature  t is,  substituting  these 
values  for  their  respective  letters  in  the  formula, 

2 X .0951  X 80  + 7.8  X 1 X 80  + 3i  X .1138  X 780 

2 X .0951  + 7.8  X 1 + 3i  X .1138 
15.216  + 624  + 288.483  __  927.699 
.1902  + 7.8  + .36985  ~ 8.36005  — 11U'y?  * 

In  substituting  the  numerical  values,  the  signs  of  multi- 
plication are,  of  course,  written  in  their  proper  places;  all 
the  multiplications  are  performed  before  adding,  according 
to  the  rule  previously  given. 


100 


FORMULAS. 


The  reader  should  now  be  able  to  apply  any  formula 
involving  only  algebraic  expressions  that  he  may  meet  with, 
not  requiring  the  use  of  logarithms  for  their  solution.  We 
will,  however,  call  his  attention  to  one  or  two  other  facts 
which  he  may  have  forgotten. 

160 

Expressions  similar  to  — sometimes  occur,  the  heavy  line 
"25 

indicating  that  160  is  to  be  divided  by  the  quotient  obtained 
by  dividing  660  by  25.  If  both  lines  were  light  it  would  be 

impossible  to  tell  whether  160  was  to  be  divided  by  or 
160 

whether  — - was  to  be  divided  by  25.  If  this  latter  result 
660 

160 

were  desired,  the  expression  would  be  written  In  every 

case  the  heavy  line  indicates  that  all  above  it  is  to  be  divided 
by  all  below  it. 

160 

In  an  expression  like  the  following, — , the  heavy  line 

tn  66U 


is  not  necessary,  since  it  is  impossible  to  mistake  the  opera- 

660 

tion  that  is  required  to  be  performed.  But,  since  7 + -^r 

175  -f  660  . , ...  , 175  + 660  „ „ , 660  , 

= — ~ — , if  we  substitute  — ■== — for  7 + — , the  heavy 

25  ZD  ZD 

line  becomes  necessary  in  order  to  make  the  resulting  expres- 
sion clear.  Thus, 

160  160  160 

660  “ 175  + 660  “ 835' 

7 + 25  25  25 


Fractional  exponents  are  sometimes  used  instead  of  the 
radical  sign.  That  is,  instead  of  ind' mating  the  square,  cube, 
fourth  root,  etc.  of  some  quantity,  as  37  by  j/  37,  ^ 37,  W, 
etc.  these  roots  are  indicated  by  37*  37*  37*,  etc.  Should 
the  numerator  of  the  fractional  exponent  be  some  quantity 
other  than  1,  this  quantity,  whatever  it  may  be,  indicates 
that  the  quantity  affected  by  the  exponent  is  to  be  raised  to 
the  power  indicated  by  the  numerator;  the  denominator  is 


FORMULAS. 


101 


always  the  index  of  the  root.  Hence^jnstead  of  expressing 
the  cube  root  of  the  square  of  37  as  $ 372,  it  may  be  expressed 
37^,  the  denominator  being  the  index  of  the  root;  in  other 
words,  ^ S72  = 37 Likewise,  (1  + a2b )3  may  also  be 

written  (1  + a26)*,  a much  simpler  expression. 

We  will  now  give  several  examples  showing  how  to  apply 
some  of  the  more  difficult  formulas  that  the  reader  may 
encounter. 

The  area  of  any  segment  of  a circle  that  is  less  than  (or 
equal  to)  a semicircle  is  expressed  by  the  formula. 


A 


7T  r2E 
~360 


in  which  A = area  of  segment; 

7T  = 3.1416; 
r = radius; 

E = angle  obtained  by  drawing  lines  from  the 
center  to  the  extremities  of  arc  of  segment; 
c = chord  of  segment; 
h — height  of  segment. 

Example.— What  is  the  area  of  a segment  whose  chord  is 
10  in.  long,  angle  subtended  by  chord  is  83.46°,  radius  is 
7.5  in.,  and  height  of  segment  is  1.91  in.  ? 

Solution.— Applying  the  formula  just  given, 


A = 


r r2E 


c I 

-2(r“ 


7i)  = 


3.1416X7.52X83.46 


360  2 v 360 

= 40.968  — 27.95  = 13.018  sq.  in.,  nearly. 


j (7.5-1.91) 


The  area  of  any  triangle  may  be  found  by  means  of  the 
following  formula,  in  which  A = the  area,  and  a,  5,  and  c 
represent  the  lengths  of  the  sides: 


A 


4 a H2^)’ 


Example.— What  is  the  area  of  a triangle  whose  sides  are 
21  ft.,  46  ft.,  and  50  ft.  long  ? 

Solution.— In  order  to  apply  the  formula,  suppose  we  let 
a represent  the  side  that  is  21  ft.  long;  b,  the  side  that  is  50  ft. 
long;  and  c,  the  side  that  is  46  ft.  long.  Then,  substituting 
in  the  formula, 


102 


FORMULAS. 


-i  VH^-)'  - 
-IV* 


,441- 


441  + 2,500- 
100 


- 2,11 6^  2 


= 25 1/441  - 8.252  = 25  l/441  — 68.0625  = 25  V 372.9375 


= 25  X 19.312  = 482.8  sq.  ft.,  nearly. 

The  above  operations  have  been  extended  much  further 
than  was  necessary;  this  was  done  in  order  to  show  the 
reader  every  step  of  the  process. 

The  Rankine-Gordon  formula  for  determining  the  least 
load  in  pounds  that  will  cause  a long  column  to  break  is 


P = 


S A 

1+?Z 


in  which  P = load  (pressure)  in  lb.;  S = ultimate  strength 
(in  lb.  per  sq.  in.)  of  material  composing  column;  A = area 
of  cross-section  of  column  in  sq.  in.;  q = a factor  (multiplier) 
whose  value  depends  on  the  shape  of  the  ends  of  the  column 
and  on  the  material  composing  the  column;  l = length  of  the 
column  in  in.;  G = least  radius  of  gyration  of  cross-section 
of  column. 

The  values  of  S,  q,  and  G2  are  all  given  in  printed  tables 
on  pages  151,  153,  and  156. 

Example. — What  is  the  least  load  that  will  break  a hollow 
steel  column  whose  outside  diameter  is  14  in.,  inside  diam- 
eter 11  in.,  length  20  ft.,  and  whose  ends  are  flat? 


Solution.— For  steel,  S = 150,000,  and  q = — for  flat- 

25,000 

ended  steel  columns;  A,  the  area  of  the  cross-section,  = 
.7854(di2-  d22),  cli  and  do  being  the  outside  and  inside  diam- 
eters, respectively;  l = 20  X 12  = 240 in.;  and  G2  = -r  j~ 


Substituting  +ese  values  in  the  formula, 

SA  150,000  X .7854(142-112)  _ 


P = — - 


1 + qG* 


i+  1 : ^ 

x 25,000  A 142  + ll2 


150.000  X 58.905 
1 + .1163 


8,835.750 

1.1163 


16 

7,915,211  lb. 


INVOLUTION  AND  EVOLUTION. 


103 


INVOLUTION  AND  EVOLUTION. 

By  means  of  the  following  table  the  square,  cube,  square 
root,  cube  root,  and  reciprocal  of  any  number  may  be 
obtained  correct  always  to  five  significant  figures,  and  in  the 
majority  of  cases  correct  to  six  significant  figures. 

In  any  number,  the  figures  beginning  with  the  first  digit  * 
at  the  left  and  ending  with  the  last  digit  at  the  right,  are 
called  the  significant  figures  of  the  number.  Thus,  the  num- 
ber 405,800  has  the  four  significant  figures  4,  0,  5,  8;  and  the 
number  .000090067  has  the  five  significant  figures  9,  0,  0,  6, 
and  7. 

The  part  of  a number  consisting  of  its  significant  figures 
is  called  the  significant  part  of  the  number.  Thus,  in  the 
number  28,070,  the  significant  part  is  2807;  in  the  number 
.00812,  the  significant  part  is  812;  and  in  the  number  170.3,  the 
significant  part  is  1703. 

In  speaking  of  the  significant  figures  or  of  the  significant 
part  of  a number,  the  figures  are  considered,  in  their  proper 
order,  from  the  first  digit  at  the  left  to  the  last  digit  at  the 
right,  but  no  attention  is  paid  to  the  position  of  the  decimal 
point.  Hence,  all  numbers  that  differ  only  in  the  position  of 
the  decimal  point  have  the  same  significant  part.  For  example, 
.002103,  21.03,  21,030,  and  210,300  have  the  same  significant 
figures  2,  1,  0,  and  3,  and  the  same  significant  part  2103. 

The  integral  part  of  a number  is  the  part  to  the  left  of  the 
decimal  point. 

It  will  be  more  convenient  to  explain  first  how  to  use  the 
table  for  finding  square  and  cube  rootg. 


SQUARE  ROOT. 

First  point  off  the  given  number  into  periods  of  two  figures 
each,  beginning  with  the  decimal  point  and  proceeding  to 
the  left  and  right.  The  following  numbers  are  thus  pointed 
off:  12703,  1'27'03;  12.703,  12.70'30;  220000,  22W00;  .000442, 
.00W42. 


A cipher  is  not  a digit. 


104 


SQUARE  ROOT. 


Haying  pointed  off  the  number,  move  the  decimal  point 
bo  that  it  will  fall  between  the  first  and  second  periods  of  the 
significant  part  of  the  number.  In  the  above  numbers,  the 
decimal  point  will  be  placed  thus:  1.2703, 12.703,  22,  4.42. 

If  the  number  has  but  three  (or  less)  significant  figures, 
find  the  significant  part  of  the  number  in  the  column  headed 
n;  the  square  root  will  be  found  in  the  column  headed  j/n 
or  |/  10  n,  according  to  whether  the  part  to  the  left  of  the 
decimal  point  contains  one  figure  or  two  figures.  Thus,  j/ 4.42 
= 2.1024,  and  i/  22  = 4/ 10  X 2.20  = 4.6904.  The  decimal  point 
is  located  in  all  cases  by  reference  to  the  original  number 
after  pointing  off  into  periods. 

There  will  be  as  many  figures  in  the  root  preceding  the  decimal 
point  as  there  are  periods  preceding  the  decimal  point  in  the 
given  number;  if  the  number  is  entirely  decimal , the  root  is 
entirely  decimal , and  there  will  be  as  many  ciphers  following  the 
decimal  point  in  the  root  as  there  are  cipher  periods  following 
the  decimal  point  in  the  given  number. 

Applying  this  rule,  )/ 220000  = 469.04  and  ]/. 000442  = 
.021024. 

The  operation  when  the  given  number  has  more  than 
three  significant  figures  is  best  explained  by  an  example. 

Example.— (a)  4/ 3J.416  = ? (6)  1/ 2342.9  = ? 

Solution.— (a)  Since  the  first  period  contains  but  one 
figure,  there  is  no  need  of  moving  the  decimal  point.  Look 
in  the  column  headed  n2  and  find  two  consecutive  numbers, 
one  a little  greater  and  the  other  a little  less  than  the  given 
number ; in  the  present  case,  3.1684  = 1.782  and  3.1329  = 1.772. 
The  first  three  figures  of  the  root  are  therefore  177.  Find  the 
difference  between  the  two  numbers  between  which  the 
given  number  falls,  and  the  difference  between  the  smaller 
number  and  the  given  number ; divide  the  second  difference 
by  the  first  difference,  carrying  the  quotient  to  three  decimal 
places  and  increasing  the  second  figure  by  1 if  the  third 
is  6 or  a greater  digit.  The  two  figures  of  the  quotient 
thus  determined  will  be  the  fourth  and  fifth  figures  of  the 
root.  In  the  present  example,  dropping  decimal  points 
in  the  remainders,  3.1684  — 3.1329  = 355,  the  first  difference; 


INVOLUTION  AND  EVOLUTION. 


105 


3.1416  — 3 1329  = 87,  the  second  difference;  87  h-  355  = .245+, 
or  .25.  Hence,  V 3.1416  = 1.7725. 

(6)  i/*2342.9  = ? Pointing  off  into  periods  we  get  23'42.90; 
moving  the  decimal  point  we  get  23.4290;  the  first  three 
figures  of  the  root  are  484;  the  first  difference  is  23.5225  — 
23.4256  = 969;  the  second  difference  is  23.4290  — 23.4256  = 34; 
34  -f-  969  = .035+,  or  .04.  Hence,  = 48.404. 


CUBE  ROOT. 

The  cube  root  of  a number  is  found  in  the  same  manner 
as  the  square  root,  except  the  given  number  is  pointed  off 
into  periods  of  three  figures  each.  The  following  numbers 
would  be  pointed  off  thus:  3141.6,  3'141.6;  67296428,  67'296'428; 
601426.314,  60P426.314;  .0000000217,  .000'000'021'700. 

Having  pointed  off,  move  the  decimal  point  so  that  it  will 
fall  between  the  first  and  second  periods  of  the  significant 
part  of  the  number,  as  in  square  root.  In  the  above  num- 
bers the  decimal  point  will  be  placed  thus:  3.1416,  67.296428, 
601.426314,  and  21.7. 

If  the  given  number  has  but  three  (or  less)  significant 
figures,  find  the  significant  part  of  the  number  in  the 
column  headed  w;  the  cube  root  will  be  found  in  the  column 
headed  f/  n , i/l0  n,  or  ]/l00  n,  according  to  whether  one, 
two,  or  three  figures  precede  the  decimal  point  after  it  has 
been  moved.  Thus,  the  cube  root  of  21.7  will  be  found  oppo- 
site 2.17,  in  column  headed  ^ 10  n,  while  the  cube  root  of  2.17 
would  be  found  in  the  column  headed  n,  and  the  cube 
root  of  217  in  the  column  headed  ^ 100  n,  all  on  the  same 
line.  If  the  given  number  contains  more  than  three  sig- 
nificant figures,  proceed  exactly  as  described  for  square  root 
except  that  the  column  headed  n3  is  used. 

Example.— (a)  ^0000062417  = ? (5)  1/ 50932676  = ? 

Solution.— (a)  Pointing  off  into  periods,  we  get 
000'006'241'700;  moving  the  decimal  point,  we  get  6.2417. 
The  number  falls  between  6.22950  = 1.843  and  6.33163  = 1.85s; 
the  first  difference  = 10213;  the  second  difference  is 


106 


SQUARES  AND  CUBES. 


6.24170  - 6.22950  = 1220;  1220  -4- 10213  = .119+,  or  .12,  the  fourth 
and  fifth  figures  of  the  root.  The  decimal  point  is  located  by 
the  rule  previously  given;  hence,  1^.0000(162417  = .018412. 

(6)  ^ 50932676  = ? As  the  number  contains  more  than 

six  significant  figures,  reduce  it  to  six  significant  figures  by 
replacing  all  after  the  sixth  figure  with  ciphers,  increasing 
the  sixth  figure  by  1 when  the  seventh  is  5 or  a greater 
digit.  In  other  words,  the  first  five  figures  of  ^ 50932700  and 
of  i^ 50932676  are  the  same.  Pointing  off  into  periods,  we  get 
50'932'700;  moving  the  decimal  point,  we  get  50.9327,  which 
falls  between  50.6530  = 3.703  and  51.0648  = 3.713;  the  first 
difference  is  4118;  the  second  difference  is  2797;  2797  -4-  4118 
= .679+,  or  .68.  The  integral  part  of  the  root  evidently  con- 
tains three  figures;  hence,  ^ 50932676  = 370.68,  correct  to  five 
figures. 


SQUARES  AND  CUBES. 

If  the  given  number  contains  but  three  (or  less)  signifi- 
cant figures,  the  square  or  cube  is  found  in  the  column 
headed  w2  or  n 3,  opposite  the  given  number  in  the  column 
headed  n.  If  the  given  number  contains  more  than  three 
significant  figures,  proceed  in  a manner  similar  to  that 
described  for  extracting  roots.  To  square  a number,  place  the 
decimal  point  between  the  first  and  second  significant  figures 
and  find  in  the  column  headed  \/ n or  ]/ 10  n two  consecu- 
tive numbers,  one  of  which  shall  be  a little  greater  and  the 
other  a little  less  than  the  given  number.  The  remainder  of 
the  work  is  exactly  as  heretofore  described.  To  locate  the 
decimal  point,  employ  the  principle  that  the  square  of  any 
number  contains  either  twice  as  many  figures  as  the  num- 
ber squared  or  twice  as  many  less  one.  If  the  column  headed 
■j/ 10  n is  used,  the  square  will  contain  twice  as  many  figures, 
while  if  the  column  headed  \/ n is  used,  the  square  will 
contain  twice  as  many  figures  as  the  number  squared,  less 
one.  If  the  number  contains  an  integral  part,  the  principle 
is  applied  to  the  integral  part  only;  if  the  number  is  wholly 
decimal,  there' will  be  twice  as  many  ciphers  following  the 


INVOLUTION  AND  EVOLUTION. 


107 


decimal  in  the  square  or  twice  as  many  plus  one  as  in 
the  number  squared,  depending  on  whether  |/l0n  or  j/n 
column  is  used.  For  example,  273.4‘22  will  contain  five  figures 
in  the  integral  part;  4516.22  will  contain  eight  figures  in  the 
integral  part,  all  after  the  fifth  being  denoted  by  ciphers; 
.00294532  will  have  five  ciphers  following  the  decimal  point; 
.0524362  will  have  two  ciphers  following  the  decimal  point. 

Example.— (a)  273.422  = ? (6)  .052436 2 = ? 

Solution.— ( a)  Placing  the  decimal  point  between  the  first 
and  second  significant  figures,  the  result  is  2.7342;  this  number 
occurs  between  2.73313  = V 7.47  and  2.73496  = j/  7.48  in  the 
column  headed  y n.  The  first  difference  is  2.73496  — 2.73313 
= 183;  the  second  difference  is  2.73420  — 2.73313  = 107;  and 
107  -r-  183  = .584+,  or  .58.  Hence,  273.42 2 = 74,758,  correct  to 
five  significant  figures. 

(6)  Shifting  the  decimal  point  to  between  the  first  and 
second  significant  figures,  we  get  the  number  5.2436,  which 
falls  between  5.23450  = i/27.4  and  5.24404  = i/  27.5.  The  first 
difference  is  954;  the  second  difference  is  910;  910  -j-  954  = 
.953-+  or  .95.  Hence,  .052436 2 = .0027495,  to  five  significant 
figures. 

A number  is  cubed  in  exactly  the  same  manner,  using  the 
column  headed  ^ n,  ^ 10  n,  or  ^ 100  ?i,  according  to  whether 
the  first  period  of  the  significant  part  of  the  number  contains 
one,  two,  or  three  figures,  respectively.  If  the  number  con- 
tains an  integral  part,  the  number  of  figures  in  the  integral 
part  of  the  cube  will  be  three  times  as  many  as  in  the  given 
number  if  column  headed  $ 100  n is  used;  it  will  be  three 
times  as  many  less  1 if  the  column  headed  $ 10 n is  used; 
and  it  will  be  three  times  as  many  less  2 if  the  column 
headed  n is  used.  If  the  given  number  is  wholly  decimal 
the  cube  will  have  either  three  times,  three  times  plus 
one,  or  three  times  plus  two,  as  many  ciphers  following  the 
decimal  as  there  are  ciphers  following  the  decimal  point  in 
the  given  number. 

Example.— (a)  129.6843=?  (6)  .764422  = ?.  (c)j  .032425* 

= ? 

Solution.— (a)  Placing  the  decimal  point  between  the 


108 


RECIPROCALS. 


first  and  second  significant  figures,  the  number  1.29684  is  found 
between  1.29664  = fziS  and  1.29862  = f '~2A9.  The  first 
difference  is  198;  the  second  difference  is  20;  and  20  -s-  198 
= .101+,  or  .10.  Hence,  the  first  five  significant  figures  are 
21810;  the  number  of  figures  in  the  integral  part  of  the  cube  is 
3 X 3 — 2 = 7;  and  129.684 3 = 2,181,000,  correct  to  five  sig- 
nificant figures. 

(ft)  7.64420  occurs  between  7.64032  = 1^446  and  7.64603  == 
^ 447.  The  first  difference  is  571;  the  second  difference  is 
388;  and  388  -5-  571  = .679+,  or  .68.  Hence,  the  first  five  signifi- 
cant figures  are  44668;  the  number  of  ciphers  following  the 
decimal  point  is  3 X 0 = 0;  and  .764423  = .44668,  correct  to  five 
significant  figures. 

(c)  3.2425  falls  between  3.24278  = ^ 34.1  and  3.23961  = 
1^34.0.  The  first  difference  is  317;  the  second  difference  is 
289;  289  -4-  317  = .911+,  or  .91.  Hence,  the  first  five  significant 
figures  are  34091;  the  number  of  ciphers  following  the  decimal 
point  is  3 X 1 + 1 = 4;  and  .032425 3 = .000034091,  correct  to 
five  significant  figures. 


RECIPROCALS. 

The  reciprocal  of  a number  is  1 divided  by  the  number. 
By  using  reciprocals,  division  is  changed  into  multiplication, 

since  a^-ft  = ^ = a X The  table  gives  the  reciprocals  of 

all  numbers  expressed  with  three  significant  figures  to  six 
significant  figures.  By  proceeding  in  a manner  similar  to 
that  just  described  for  powers  and  roots,  tne  reciprocal  of  any 
number  correct  to  five  significant  ngures  may  be  obtained. 
The  decimal  point  in  the  result  may  be  located  as  follows* 
If  the  given  number  has  an  integral  part,  the  number  of 
ciphers  following  the  decimal  point  in  the  reciprocal  will  be 
one  less  than  the  number  of  figures  in  the  integral  part  of  the 
given  number;  and  if  the  given  number  is  entirely  decimal, 
the  number  of  figures  in  the  integral  part  of  the  reciprocal 
will  be  one  greater  than  the  number  of  ciphers  following  the 
decimal  point  in  the  given  number.  For  example,  the  recip- 
rocal of  3370  = .000296736  and  of  .00348  = 287.356. 


INVOLUTION  AND  EVOLUTION. 


109 


When  the  number  whose  reciprocal  is  desired  contains 
more  than  three  significant  figures,  express  the  number  to 
six  significant  figures  (adding  ciphers,  if  necessary,  to  make 
six  figures)  and  find  between  what  two  numbers  in  the 

column  headed  ^ the  significant  figures  of  the  given  number 

falls;  then  proceed  exactly  as  previously  described  to  deter- 
mine the  fourth  and  fifth  figures. 

Example.—  (a)  The  reciprocal  of  379.426  =?  ( b ) -nnn~L00-  = ? 

Solution. — (a)  .379426  falls  between  .378788  = — ^ and 

.380228  = -i- . The  first  difference  is  380228  — 378788  = 1440; 
Abo 

the  second  difference  is  380228  — 379426  = 802;  802  1440 

— .557,  or  .56.  Hence,  the  first  five  significant  figures  are 
26356,  and  the  reciprocal  of  379.426  is  .0026356,  to  five  sig- 
nificant figures. 

(6)  .469200  falls  between  .469484  = —^3  an(*  -467290  = 

The  first  difference  i&  2194;  the  second  difference  is  284;  284 

-7-  2194  = .129+,  or  .13.  Hence,  — "1-  ■ = 2131.3,  correct  to 

.0004692 

five  significant  figures. 


110  POWERS,  ROOTS,  AND  RECIPROCALS. 


yllOn 

1 

n 

n 

n2 

n 3 

Vn 

A/lU  n 

^100 

1.01 

1.0201 

1 03030 

1.00499 

3.17805 

1.00332 

2.16159 

4.65701 

.990099 

1.02 

1.0404 

1.06121 

1.00995 

3.19374 

1.00662 

2.16870 

4.67233 

.980392 

1.03 

1.0609 

1.09273 

1.01489 

3.20936 

1.00990 

2.17577 

4.68755 

.970874 

1.04 

1.0816 

1.12486 

1.01980 

3.22490 

1.01316 

2.18278 

4.70267 

.961539 

1.05 

1.1025 

1.15763 

1.02470 

3.24037 

1.01640 

2.18976 

4.71769 

.952381 

1.06 

1.1236 

1.19102 

1.02956 

3.25576 

1.01961 

2.19669 

4.73262 

.943396 

1.07 

1 1449 

1.22504 

1.03441 

3.27109 

1.02281 

2.20358 

4.74746 

.934579 

1.08 

1.1664 

1.25971 

1.03923 

3.28634 

1.02599 

2.21042 

4.76220 

.925926 

1.09 

1.1881 

1.29503 

1.04403 

3.30151 

1.02914 

2.21722 

4.77686 

.917431 

1.10 

1.2100 

1.33100 

1.04881 

3.31662 

1.03228 

2.22398 

4.79142 

.909091 

1.11 

1.2321 

1 36763 

1.05357 

3.33167 

1.03540 

2.23070 

4.80590 

.900901 

1.12 

1.2544 

1.40493 

1.05830 

3.34664 

1.03850 

2,23738 

4.82028 

.892857 

1.13 

1.2769 

1.44290 

1.06301 

3.36155 

1.04158 

2.24402 

4.83459 

.88*956 

1.14 

1.2996 

1.48154 

1.06771 

3.37639 

1.04464 

2.25062 

4.84881 

.877193 

1.15 

1.3225 

1.52088 

1.07238 

3.39116 

1.04769 

2.25718 

4.86294 

.869565 

1.16 

1.3456 

1.56090 

1.07703 

3.40588 

1.05072 

2.26370 

4.87700 

.862069 

1.17 

1.3689 

1.60161 

1.08167 

3.42053 

1.05373 

2.27019 

4.89097 

.854701 

1.18 

1 .3924 

1.64303 

1.08628 

3.43511 

1.05672 

2.27664 

4.90487 

.847458 

1.19 

1.4161 

1.68516 

1.09087 

3.44964 

1.05970 

2.28305 

4.91868 

.840336 

1.20 

1.4400 

1.72800 

1.09545 

3.46410 

1.06266 

2.28943 

4.93242 

.833335 

1.21 

1.4641 

1.77156 

1.10000 

3.47851 

1.06560 

2.29577 

4.94609 

.826446 

1.22 

1.4884 

1.81585 

1.10454 

3.49285 

1.06853 

2.30208 

4.95968 

.819672 

1.23 

1.5129 

1.86087 

1.10905 

3.50714 

1.07144 

2.30835 

4.97319 

.813008 

1.24 

1.5376 

1.90662 

1.11355 

3.52136 

1.07434 

2.31459 

4.98663 

.806452 

1.25 

1.5625 

1.95313 

1.11803 

3.53553 

al. 07722 

2.32080 

5.00000 

.800000 

jl.26 

1.5876 

2.00038 

1.12250 

3.54965 

1.08008 

2.32697 

5.01330 

.793651 

1.27 

1.6129 

2.04838 

1.12694 

3.56371 

1.08293 

2.33310 

5.02653 

.787402 

1.28 

1.6384 

2.09715 

1.13137 

3.57771 

1.08577 

2.33921 

5.03968 

,.781250 

1.29 

1.6641 

2.14669 

1.13578 

3.59166 

1.08859 

2.34529 

5.05277 

.775194 

1.30 

1.6900 

2.19700 

1.14018 

3.60555 

1.09139 

2.35134 

5.06580 

.769231 

1.31 

1.7161 

2.24809 

1.14455 

3.61939 

1.09418 

2.35735 

5.07875 

.763359 

1.32 

1.7424 

2.29997 

1.14891 

3.63318 

1.09696 

2.36333 

5.09164 

.757576 

1.33 

1.7689 

2.35264 

1.15326 

3.64692 

1.09972 

2.36928 

5.10447 

.751880 

1.34 

1.7956 

2.40610 

1.15758 

3.66060 

1.10247 

2.37521 

5.11723 

.746269 

1.35 

1.8225 

2.46038 

1.16190 

3.67423 

1.10521 

2.38110 

5.12993 

.740741 

1.36 

1.8496 

.2.51546 

1.16619 

3.68782 

1.10793 

2.38696 

5.14256 

.735294 

1.37 

1.8769 

2.57135 

1.17047 

3.70135 

1.11064 

2.39280 

5.15514 

.729927 

1.38 

1.9044 

2.62807 

1.17473 

3.71484 

1.11334 

2.39861 

5.16765 

.724638 

1.39 

1.9321 

2.68562 

1.17898 

3.72827 

1.11602 

2.40439 

5.18010 

.719425 

1.40 

1.9600 

2.74400 

1.18322 

3.74166 

1.11869 

2.41014 

5.19249 

.714286 

1.41 

1.9881 

2.80322 

1.18743 

3.75500 

1.12135 

2.41587 

5.20483 

.709220 

1.42 

2.0164 

2.86329 

1.19164 

3.76829 

1.12399 

2.42156 

5.21710 

.704225 

1.43 

2.0449 

2.92421 

1.19583 

3.78153 

1.12662 

2.42724 

5.22932 

.699301 

1.44 

2.0736 

2.98598 

1.20000 

3 79473 

1.12924 

2.43288 

5.24148 

.694444 

1.45 

2.1025 

3.04863 

1.20416 

3.80789 

1.13185 

2.43850 

5.25359 

.689655 

1.46 

2.1316 

3.11214 

1.20830 

3.82099 

1.13445 

2.44409 

5.26564 

.684932 

1.47 

2.1609 

3.17652 

1.21244 

3.83406 

1.13703 

2.44966 

5.27763 

.680272 

1.48 

2.1904 

3.24179 

1.21655 

3.84708 

1.13960 

2.45520 

5.28957 

.675676 

1.49 

2.2201 

3.30795 

1.22066 

3.86005 

1.14216 

2.46072 

5.30146 

.671141 

1.50 

2.2500 

3.37500 

1.22474 

3.87298 

1.14471 

2.46621 

5.31329 

.666667 

POWERS,  ROOTS,  AND  RECIPROCALS.  Ill 


n 

7*3 

7l3 

yin 

VlO  11 

fin 

^10  11 

1 

n 

^100?* 

1.51 

2.2801 

3.44295 

1.22882 

3.88587 

1.14725 

2.47168 

5.32507 

.662252 

1.52 

2.3104 

3.51181 

1.23288 

3.89872 

1.14978 

2.47713 

5.33680 

.657895 

1.53 

2.3409 

3.58158 

1.23693 

3.91152 

1.15230 

2.48255 

5.34848 

.653595 

1.54 

2.3716 

3. §5226 

1.24097 

3.92428 

1.15480 

2.48794 

5.36011 

.649351 

1.55 

2.4025 

3.72388 

1.24499 

3.93700 

1.15729 

2.49332 

5.37169 

.645161 

1.56 

2.4336 

3.79642 

1.24900 

3.94968 

1.15978 

2.49866 

5.38321 

.641026 

1.57 

2.4649 

3.86989 

1.25300 

3.96232 

1.16225 

2.50399 

5.39469 

.636943 

1.58 

2.4964 

3.94431 

1 .25698 

3.97492 

1.16471 

2.50930 

5.40612 

.632911 

1.59 

2.5281 

4.01968 

1.26095 

3.98748 

1.16717 

2.51458 

5.41750 

.628931 

1.60 

2.5600 

4.09600 

1.26491 

4.00000 

1.16961 

2.51984 

5.42884 

.625000 

1.61 

2.5921 

4.17328 

1.26886 

4.01248 

1.17204 

2.52508 

5.44012 

.621118 

1.62 

2.6244 

4.25153,1 

1.27279 

4.02492 

1.17446 

2.53030 

5.45136 

.617284 

1.63 

2.6569 

4.33075 

1.27671 

4.03733 

1.17687 

2.53549 

5.46256 

.613497 

1.64 

2.6896 

4.41094 

1.28062 

4.04969 

1.17927 

2.54067 

5.47370 

.609756 

1.65 

2.7225 

4.49213 

1.28452 

4.06202 

1.18167 

2.54582 

5.48481 

.606061 

1.66 

2.7556 

4.57430 

1.28841 

4.07431 

1.18405 

2.55095 

5.49586 

.602410 

1.67 

2.7889 

4.65746 

1.29228 

4.08656 

1.18642 

2.55607 

5.50688 

.598802 

1.68 

2.8224 

4.74163 

1.29615 

4.09878 

1.18878 

2.56116 

5.51785 

.595238 

1.69 

2.8561 

4.82681 

1.30000 

4.11096 

1.19114 

2.56623 

5.52877 

.591716 

1.70 

2.8900 

4.91300 

1.30384 

4.12311 

1.19348 

2.57128 

5.53966 

.588235 

1.71 

2.9241 

5.00021 

1.30767 

4.13521 

1.19582 

2.57631 

5.55050 

.584795 

1.72 

2.9584 

5.08845 

1.31149 

4.14729 

1.19815 

2.58133 

5.56130 

.581395 

1.73 

2.9929 

5.17772 

1.31529 

4.15933 

1.20046 

2.58632 

5.57205 

.578035 

1.74 

3.0276 

5.26802 

1.31909 

4.17133 

1.20277 

2.59129 

5.58277 

.574713 

1.75 

3.0625 

5.35938 

1.32288 

4.18330 

1.20507 

2.59625 

5.59344 

.571429 

1.76 

3.0976 

5.45178 

1 .32665 

4.19524 

1.20736 

2.60118 

5.60408 

.568182 

1.77 

3.1329 

5.54523 

1.33041 

4.20714 

1.20964 

2.60610 

5.61467 

.564972 

1.78 

3.1684’ 

5.63975 

1.33417 

4.21900 

1.21192 

2.61100 

5.62523 

.561798 

1.79 

3.2041" 

5.73534 

1.33791 

4.23084 

1.21418 

2.61588 

5.63574 

.558659 

1.80 

3.2400 

5.83200 

1.34164 

4.24264 

1.21644 

2.62074 

5.64622 

.555556 

1.81 

3.2761 

5.92974 

1.34536 

4.25441 

1.21869 

2.62558 

5.65665 

.552486 

1.82 

3.3124 

6.02857 

1.34907 

4.26615 

1.22093 

2.63041 

5.66705 

.549451 

1.83 

3.3489 

6.12849 

1.35277 

4.27785 

1.22316 

2.63522 

5.67741 

.546448 

1.84 

3.3856 

6.22950 

1.35647 

4.28952 

1.22539 

2.64001 

5.68773 

.543478 

1.85 

3.4225 

6.33163 

1.36015 

4.30116 

1.22760 

2.64479 

5.69802 

.540541 

1.86 

3.4596 

6.43486 

1.36382 

4.31277 

1.22981 

2.64954 

5.70827 

.537634 

1.87 

3.4969 

6.53920 

1.36748 

4.32435 

1 23201 

2.65428 

5.71848 

.534759 

1.88 

3.5344 

6.64467 

1.37113 

4.33590 

1.23420 

2.65900 

5.72865 

.531915 

1.89 

3.5721 

6.75127 

1.37477 

4.34741 

1.23639 

2.66371 

5.73879 

.529101 

1.90 

3.6100 

6.85900 

1.37840 

4.35890 

1.23856 

2.66840 

5.74890 

.526316 

1.91 

3.6481 

6.96787 

1.38203 

4.37035 

1.24073 

2.67307 

5.75897 

.523560 

1.92 

3.6864 

7.07789 

1.38564 

4.38178 

1.24289 

2.67773 

5.76900 

.520833 

1.93 

3.7249 

7.18906 

1.38924 

4.39318 

1.24505 

2.68237 

5.77900 

.518135 

1.94 

3.7636 

7.30138 

1.39284 

4.40454 

1.24719 

2.68700 

5.78896 

.515464 

1.95 

3.8025 

7.41488 

1.39642 

4.41588 

1.24933 

2.69161 

5.79889 

.512821 

1.96 

3.8416 

7.52954 

1.40000 

4.42719 

1.25146 

2.69620 

5.80879 

.510204 

1.97 

3.8809 

7.64537 

1.40357 

4.43847 

1.25359 

2.70078 

5.81865 

.507614 

1.98 

3.9204 

7.76239 

1.40712 

4.44972 

1.25571 

2.70534 

5.82848 

.505051 

1.99 

3.9601 

7.88060 

1.41067 

4.46094 

1.25782 

2.70989 

5.83827 

.502513 

2.00 

4.0000 

8.00000 

1.41421 

4.47214 

1.25992 

2.71442 

5.84804 

.500000 

112  POWERS,  ROOTS,  AND  RECIPROCALS. 


n 

7l2 

n 3 

Vw 

A/10 ~n 

'iln 

ylio  n 

^T00n 

1 

n 

2.01 

4.0401 

8.12060 

1.41774 

4.48330 

1.26202 

2.71893 

5.85777 

.497512 

2.02 

4.0804 

8.24241 

1.42127 

4.49444 

1.26411 

! 2.72343 

5.86746 

.495050 

2.03 

4.1209 

8.36543 

1.42478 

4.50555 

1.26619 

2.72792 

5.87713 

.492611 

2.04 

4.1616 

8.48966 

1.42829 

4.51664 

1.26827 

2.73239 

5.88677 

.490196 

2.05 

4.2025 

8.61513 

1.43178 

4.52769 

1.27033 

2.73685 

5.89637 

.487805 

2.06 

4.2436 

8.74182 

1.43527 

4.53872 

1.27240 

2.74129 

5.90594 

.485437 

2.07 

4.2849 

8.86974 

1.43875 

4.54973 

1.27445 

2.74572 

5.91548 

.483092 

2.08 

4.3264 

8.99891 

1.44222 

4.56070 

1.27650 

2.75014 

5.92499 

.480769 

2.09 

4.3681 

9.12933 

1.44568 

4.57165 

1.27854 

2.75454 

5.93447 

.478469 

2.10 

4.4100 

9.26100 

1.44914 

4.58258 

1.28058 

2.75893 

5.94392 

.476191 

2.11 

4.4521 

9.39393 

1.45258 

4.59347 

1 .28261 

2.76330 

5.95334 

.473934 

2.12 

4.4944 

9.52813 

1.45602 

4.60435 

1.28463 

2.76766 

5.96273 

.471698 

2.13 

4.5369 

9.66360 

1.45945 

4.61519 

1.28665 

2.77200 

5.97209 

.469484 

2.14 

4.5796 

9.80034 

1.46287 

4.62601 

1.28866 

2.77633 

5.98142 

.467290 

2.15 

4.6225 

9.93838 

1.46629 

4.63681 

1.29066 

2.78065 

5.99073 

.465116 

2.16 

4.6656 

10.0777 

1.46969 

4.64758 

1.29266 

2.78495 

6.00000 

.462963 

2.17 

4.7089 

10.2183 

1.47309 

4.65833 

1.29465 

2.78924 

6.00925 

.460830 

2.18 

4.7524 

10.3602 

1.47648 

4.66905 

1.29664 

2.79352 

6.01846 

.458716 

2.19 

4.7961 

10.5035 

1.47986 

4.67974 

1.29862 

2.79779 

6.02765 

.456621 

2.20 

4.8400 

10.6480 

1.48324 

4.69042 

1.30059 

2.80204 

6.03681 

.454546 

2.21 

4.8841 

10.7939 

1.48661 

4.70106 

1.30256. 

2.80628 

6.04594 

.452489 

2.22 

4.9284 

10.9410 

1.48997 

4.71169 

1.30452 

2.81051 

6.05505 

.450451 

2.23 

4.9729 

11.0896 

1.49332 

4.72229 

1.30648 

2.81472 

6.06413 

.448431 

2.24 

5.0176 

11.2394 

1.49666 

4.73286 

1.30843 

2.81892 

6.07318 

.446429 

2.25 

5.0625 

11.3906 

1.50000 

4.74342 

1.31037 

2.82311 

6.08220 

.444444 

2.26 

, 5.1076 

11.5432 

1.50333 

4.75395 

1.31231 

2.82728 

6.09120 

.442478 

2.27 

5.1529 

11.6971 

1.50665 

4.76445 

1.31424 

2.83145 

6.10017 

.440529 

2.28 

5.1984 

11.8524 

1.50997 

4.77493 

1.31617 

2.83560 

6.10911 

.438597 

2.29 

5.2441 

12.0090 

1.51327 

4.78539 

1.31809 

2.83974 

6.11803 

.436681 

2.30 

5.2900 

12.1670 

1.51658 

4.79583 

1.32001 

2.84387 

6.12693 

.434783 

2.31 

5.3361 

12.3264 

1.51987 

4.80625 

1.32192 

2.84798 

6.13579 

.432900 

2.32 

5.3824 

12.4872 

1.52315 

4.81664 

1.32382 

2.85209 

6.14463 

.431035 

2.33 

5.4289 

12.6493 

1.52643 

4.82701 

1.32572 

2.85618 

6.15345 

.429185 

2.34 

5.4756 

12.8129 

1.52971 

4.83735 

1.32761 

2.86026 

6.16224 

.427350 

2.35 

5.5225 

12.9779 

1.53297 

4.84768 

1.32950 

2.86433 

6.17101 

.425532 

2.36 

5.5696 

13.1443 

1.53623 

4.85798 

1.33139 

2.86838 

6.17975 

.423729 

2.37 

5.6169 

13.3121 

1.53948 

4.86826 

1.33326 

2.87243 

6.18846 

.421941 

2.38 

5.6644 

13.4813 

1.54272 

4.87852 

1.33514 

2.87646 

6.19715 

.420168 

2.39 

5.7121 

13.6519 

1.54596 

4.88876 

1.33700 

2.88049 

6.20582 

.418410 

2.40 

5.7600 

13.8240 

1.54919 

4.89898 

1.33887 

2.88450 

6.21447 

.416667 

2.41 

5.8081 

13.9975 

1.55242 

4.90918 

1.34072 

2.88850 

6.22308 

.414938 

2.42 

5.8564 

14.1725 

1.55563 

4.91935 

1.34257 

2.89249 

6.23168 

.413223 

2.43 

5.9049 

14.3489 

1.55885 

4.92950 

1.34442 

2.89647 

6.24025 

.411523 

2.44 

5.9536 

14.5268 

1.56205 

4.93964 

1.34626 

2.90044 

6.24880 

.409836 

2.45 

6.0025 

14.7061 

1.56525 

4.94975 

1.34810 

2.90439 

6.25732 

.408163 

2.46 

6.0516 

14.8869 

1.56844 

4.95984 

1.34993 

2.90834 

6.26583 

.406504 

2.47 

6.1009 

15.0692 

1.57162 

4.96991 

1.35176 

2.91227 

6.27431 

.404858 

2.48 

6.1504 

15.2530 

1.57480 

4.97996 

1.35358 

2.91620 

6.28276 

.403226 

2.49 

6.2001 

15.4382 

1.57797 

4.98999 

1.35540 

2.92011 

6.29119 

.401606 

2.50 

6.2500 

15.6250 

1.58114 

5.00000 

1.35721 

2.92402 

6.29961 

.400000 

POWERS,  ROOTS,  AND  RECIPROCALS.  112a 


n 

7l2 

n 3 

>fn 

VlO  n 

3j— 

\?l 

<10  n 

II 

1 

i n 

2.51 

6.3001 

15.8133 

1.58430 

5.00999 

1 .35902 

2.92791 

6.30799 

.398406 

2.52 

6.3504 

16.0030 

1.58745 

5.01996 

1.36082 

1 2.93179 

6.31636 

.396825 

2.53 

6.4009 

16.1943 

1.59060 

5.02991 

1.36262 

! 2.93567 

6.32470 

.395257 

2 54 

6.4516 

16.3871 

1.59374 

5.03984 

1.36441 

2.93953 

6.33303 

.393701 

2.55 

6.5025 

16.5814 

1 .59687 

5.04975 

1.36620 

2.94338 

6.34133 

.392157 

2.56 

6.5536 

16.7772 

1.60000 

5.05964 

1.36798 

2.94721 

6.34960 

.390625 

2.57 

6.6049 

16.9746 

1.60312 

5.06952 

1.36976 

2.95106 

6.35786 

.389105 

2.58 

6.6564 

17.1735 

1 .60624 

5.07937 

1.37153 

2.95488 

6.36610 

.387597 

2.59 

6.7081 

17.3740 

1.60935 

5.08920 

1.37330 

2.95869 

6.37431 

.386100 

2.60 

6.7600 

17.5760 

1.61245 

5.09902 

1.37507 

2.96250 

6.38250 

.384615 

2.61 

6.8121 

17.7796 

1.61555 

5.10882 

1.37683 

2.96629 

6.39068 

.383142 

2.62 

6.8644 

17.9847 

1.61864 

5.11859 

1.37859 

2.97007 

6.39883 

.381679 

2.63 

6.9169 

18.1914 

1.62173 

5.12835 

1.38034 

2.97385 

6.40696 

.380228 

2.64 

6.9696 

18.3997 

1 .62481 

5.13809 

1.38208 

2.97761 

6.41507 

.378788 

2.65 

7.0225 

18.6096 

1.62788 

5.14782 

1.38383 

2.98137 

6.42316 

.377359 

2.66 

7.0756 

18.8211 

1.63095 

5.15752 

1.38557 

2.98511 

6.43123 

| .375940 

2.67 

7.1289 

19.0342 

1.63401 

5.16720 

1.38730 

2.98885 

6.43928 

.374532 

2.68 

7.1824 

19.2488 

1.63707 

5.17687 

1.38903 

2.99257 

6.44731 

.373134 

2.69 

7.2361 

19.4651 

1.64012 

5.18652 

1.39076 

2.99629 

6.45531 

.371747 

2.70 

7.2900 

19.6830 

1.64317 

5.19615 

1.39248 

3.00000 

6.46330 

.370370 

2.71 

7.3441 

19.9025 

1.64621 

5.20577 

1.39419 

3.00370 

6.47127 

.369004 

2.72 

7.3984 

20.1236 

1.64924 

5.21536 

1 .39591 

3.00739 

6.47922 

.367647 

2.73 

7.4529 

20.3464 

1.65227 

5.22494 

1.39761 

3.01107 

6.48715 

.366300 

2.74 

7.5076 

20.5708 

1.65529 

5.23450 

1.39932 

3.01474 

6.49507 

.364964 

2.75 

7.5625 

20.7969 

1.65831 

5.24404 

1.40102 

3.01841 

6.50296 

.363636 

2.76 

7.6176 

21.0246 

1.66132 

5.25357 

1.40272 

3.02206 

6.51083 

.362319 

2.77 

7.6729 

21.2539 

1.66433 

5.26308 

1.40441 

3.02571 

6.51868 

.361011 

2.78 

7.7284 

21.4850 

1.66733 

5.27257 

1.40610 

3.02934 

6.52652 

.359712 

2.79 

7.7841 

21.7176 

1.67033 

5.28205 

1.40778 

3.03297 

6.53434 

.358423 

2.80 

7.8400 

21.9520 

1.67332 

5.29150 

1.40946 

3.03659 

6.54213 

.357142 

2.81 

7.8961 

22.1880 

1.67631 

5.30094 

1.41114 

3.04020 

6.54991 

.355872 

2.82 

7.9524 

22.4258 

1.67929 

5.31037 

1.41281 

3.04380 

6.55767 

.354610 

2.83 

8.0089 

22.6652 

1.68226 

5.31977 

1.41448 

3.04740 

6.56541 

.353357 

2.84 

8.0656 

22.9063 

1.68523 

5.32917 

1,41614 

3.05098 

6.57314 

.352113 

2.85 

8.1225 

23.1491 

1.68819 

5.33854 

1.41780 

3.05456 

6.58084 

.350877 

2.86 

8.1796 

23.3937 

1.69115 

5.34790 

1.41946 

3.05813 

6.58853 

.349650 

2.87 

8.2369 

23.6399 

1.69411 

5.35724 

1.42111 

3.06169 

6.59620 

.348432 

2.88 

8.2944 

23.8879 

1.69706 

5.36656 

1.42276 

3.06524 

6.60385 

.347222 

2.89 

8.3521 

24.1376 

1.70000 

5.37587 

1.42440 

3.06878 

6.61149 

.346021 

2.90 

8.4100 

24.3890 

1.70294 

5.38516 

1.42604 

3.07232 

6.61911 

.344828 

2.91 

8.4681 

’24.6422 

1.70587 

5.39444 

1.42768 

3.07585 

6.62671 

.343643 

2.92 

8.5264 

24.8971 

1.70880 

5.40370 

1.42931 

3.07936 

6.63429 

.342466 

2.93 

8.5849 

25.1538 

1.71172 

5.41295 

1.43094 

3.08287 

6.64185 

.341297 

2.94 

8.6436 

25.4122 

1.71464 

5.42218 

1.43257 

3.08638 

6.64940 

.340136 

2.95 

8.7025 

25.6724 

1.71756 

5.43139 

1.43419 

3.08987 

6.65693 

.338983 ; 

2.96 

8.7616 

25.9343 

1.72047 

5.44059 

1.43581 

3.09336 

6.66444 

.337838 

2.97 

8.8209 

26.1981 

1.72337 

5.44977 

1.43743 

3.09684 

6.67194 

.336700 

2.98 

8.8804 

26.4636; 

1.72627 

5.45894 

1.43904 

3.10031 

6.67942 

.335571 

2.99 

8.9401 

26.7309  i 

1.72916 

5.46809 

1.44065 

3.10378 

6.68688 

.334448 

3.00 

9.0000 

27.0000 

1.73205 

5.47723 

1.44225 

3.10723 

6.69433 

.333333 

1126  POWERS,  ROOTS,  AND  RECIPROCALS. 


n 

W2 

n3 

■V5 

VlO  n 

I 

yfton 

1 s 
© 
© 

1 

n 

3.01 

9.0601 

27.2709 

1.73494 

5.48635 

1.44385 

3.11068 

6.70176 

.332226 

3.02 

9.1204 

27.5436 

1.73781 

' 5.49545 

1.44545 

3.11412 

6.70917 

.331126 

3.03 

9.1809 

27.8181 

1.74069 

5.50454 

1.44704 

3.11755 

6.71657 

.330033 

3.04 

9.2416 

28.0945 

1.74356 

5.51362 

1.44863 

3.12098 

6.72395 

.328947 

3.05 

9.3025 

28.3726 

1.74642 

5.52268 

1 45022 

3.12440 

6.73132 

.327869 

3.06 

9.3636 

28.6526 

1.74929 

5.53173 

1.45180 

3.12781 

6.73866 

.326797 

3.07 

9.4249 

28.9344 

1.75214 

5.54076 

1.45338 

3.13121 

6.74600 

.325733 

3.08 

9.4864 

29.2181 

1.75499 

5.54977 

1.45496 

3.13461 

6.75331 

i .324675 

3.09 

9.5481 

29.5036 

1.75784 

j 5.55878 

1.45653 

3.13800 

6.76061 

.323625 

3.10 

9.6100 

29.7910 

1.76068 

j 5.56776 

1.45810 

3.14138 

6.76790 

.322581 

3.11 

9.6721 

30.0802 

1.76352 

5.57674 

1.45967 

3.14475 

6.77517 

.321543 

3.12 

9.7344 

30.3713 

1.76635 

5.58570 

1.46123 

3.14812 

6.78242 

.320513 

3.13 

9.7969 

30.6643 

1.76918 

5.59464 

1.46279 

3.15148 

6.78966 

.319489 

3.14 

9.8596 

30.9591 

1.77200 

5.60357 

1.46434 

3.15484 

6.79688 

.318471 

3.15 

9.9225 

31.2559 

1.77482 

5.61249 

1.46590 

3.15818 

6.80409 

.317460 

3.16 

9.9856 

31.5545 

1.77764 

5.62139 

1.46745 

3.16152 

6.81128 

.316456 

3.17 

10.0489 

31.8550 

1.78045 

5.63028 

1.46899 

3.16485 

6.81846 

.315457 

3.18 

10.1124 

32.1574 

1.78326 

5.63915 

1.47054 

3.16817 

6.82562 

.314465 

3.19 

10.1761 

32.4618 

1 .78606 

5.64801 

1.47208 

3.17149 

6.83277 

.313480 

3.20 

10.2400 

32.7680 

1.78885 

5.65685 

1.47361 

3.17480 

6.83990 

.312500 

3.21 

10.3041 

33.0762 

1.79165 

5.66569 

1.47515 

3.17811 

6.84702 

.311527 

3.22 

10.3684 

33.3862 

1.79444 

5.67450 

1.47668 

3.18140 

6.85412 

.310559 

3.23 

10.4329 

33.6983 

1.79722 

5.68331 

1.47820 

3.18469 

6.86121 

.309598 

3.24 

10.4976 

34.0122 

1.80000 

5.69210 

1.47973 

3.18798 

6.86829 

.308642 

3.25 

10.5625 

34.3281 

1.80278 

5.70088 

1.48125 

3.19125 

6.87534 

.307692 

3.26 

10.6276 

34.6460 

1.80555 

5.70964 

1.48277 

3.19452 

6.88239 

.306749 

3.27 

10.6929 

34.9658 

1.80831 

5.71839 

1.48428 

3.19779 

6.88942 

.305810 

3.28 

10.7584 

35.2876 

1.81108 

5.72713 

1.48579 

3.20104 

6.89643 

.304878 

3.29 

10.8241 

35.6129 

1.81384 

5.73585 

1.48730 

3.20429 

6.90344 

.303951 

3.30 

10.8900 

35.9370 

1.81659 

5.74456 

1.48881 

3.20753 

6.91042 

.303030 

3.31 

10.9561 

36.2647 

1.81934 

5.75326 

1.49031 

3.21077 

6.91740 

.302115 

3.32 

11.0224 

36.5944 

1.82209 

5.76194 

1.49181 

3.21400 

6.92436 

.301205 

3.33 

11.0889 

36.9260 

1.82483 

5.77062 

1.49330 

3.21723 

6.93130 

.300300 

3.34 

11.1556 

37.2597 

1.82757 

5.77927 

1.49480 

3.22044 

6.93823 

.299401 

3.35 

11.2225 

37.5954 

1.83030 

5.78792 

1.49629 

3.22365 

6.94515 

.298508 

3.36 

11.2896 

37.9331 

1.83303 

5.79655 

1.49777 

3.22686 

6.95205 

.297619 

3.37 

11.3569 

38.2728 

1.83576 

5.80517 

1 .49926 

3.23005 

6.95894 

.296736 

3.38 

11.4244 

38.6145 

1.83848 

5.81378 

1.50074 

3.23325 

6.96582 

.295858 

3.39 

11.4921 

38.9582 

1.84120 

5.82237 

1.50222 

3.23643 

6.97268 

.294985 

3.40 

11.5600 

39.3040 

1.84391 

5.83095 

1.50369 

3.23961 

6.97953 

.294118 

3.41 

11.6281 

39.6518 

1.84662 

5.83952 

1.50517 

3.24278 

6.98637 

.293255 

3.42 

11.6964 

40.0017 

1.84932 

5.84808 

1.50664 

3.24595 

6.99319 

.292398 

3.43 

11.7649 

40.3536 

1.85203 

5.85662 

1.50810 

3.24911 

7.00000 

.291545 

3.44 

11.8336 

40.7076 

1.85472 

5.86515 

1.50957 

3.25227 

7.00680 

.290698 

3.45 

11.9025 

41.0636 

1.85742 

5.87367 

1.51103 

3.25542 

7.01358 

.289855 

3.46 

11.9716 

41.4217 

1.86011 

5.88218 

1.51249 

3.25856 

7.02035 

.289017 

3.47 

12.0409 

41.7819 

1.86279 

5.89067 

1.51394 

3.26169 

7.02711 

.288184 

3.48 

12.1104 

42.1442 

1.86548 

5.89915 

1.51540 

3.26482 

7.03385 

.287356 

3.49 

12.1801 

42.5085 

1.86815 

5.90762 

1.51685 

3.26795 

7.04058 

.286533 

3.50 

12.2500 

42.8750 

1.87083 

5.91608 

1.51829 

3.27107 

7.04730 

.285714 

POWERS,  ROOTS,  AND  RECIPROCALS.  112c 


n 

71 2 

?l3 

>I7i 

VlO  n 

«10  n 

^100  n 

1 

n 

3.51 

12.3201 

43.2436 

1.87350 

5.92453 

1.51974 

3.27418 

7.05400 

.284900 

3.52 

12.3904 

43.6142 

1.87617 

5.93296 

1.52118 

3.27729 

7.06070 

.284091 

3.53 

12.4609 

43.9870 

1.87883 

5.94138 

1.52262 

3.28039 

7.06738 

.283286 

3.54 

12.5316 

44.3619 

1.88149 

5.94979 

1.52406 

3.28348 

7.07404 

.282486 

3.55 

12.6025 

44.7389 

1.88414 

5.95819 

1.52549 

3.28657 

7.08070 

.281690 

3.56 

12.6736 

45.1180 

1.88680 

5.96657 

1.52692 

3.28965 

7.08734 

.280899 

3.57 

12.7449 

45.4993 

1.88944 

5.97495 

1.52835 

3.29273 

7.09397 

.280112 

3.58 

12.8164 

45.8827 

1.89209 

5.98331 

1.52978 

3.29580 

7.10059 

.279330 

3.59 

12.8881 

46.2683 

1.89473 

5.99166 

1.53120 

3.29887 

7.10719 

.278552 

3.60 

12.9600 

46.6560 

1.89737 

6.00000 

1.53262 

3.30193 

7.11379 

.277778 

3.61 

13.0321 

47.0459 

1.90000 

6.00833 

1.53404 

3.30498 

7.12037 

.277008 

3.62 

13.1044 

47.4379 

1.90263 

6.01664 

1.53545 

3.30803 

7.12694 

.276243 

3.63 

13.1769 

47.8321 

1.90526 

6.02495 

1.53686 

3.31107 

7.13349 

.275482 

3.64 

13.2496 

48.2285 

1.90788 

6.03324 

1.53827 

3.31411 

7.14004 

.274725 

3.65 

13.3225 

48.6271 

1.91050 

6.04152 

1.53968 

3.31714 

7.14657 

.273973 

3.66 

13.3956 

49.0279 

1.91311 

6.04979 

1.54109 

3.32017 

7.15309 

.273224 

3.67 

13.4689 

49.4309 

1.91572 

6.05805 

1.54249 

3.32319 

7.15960 

.272480 

3.68 

13.5424 

49.8360 

1.91833 

6.06630 

1.54389 

3.32621 

7.16610 

.271739 

3.69 

13.6161 

50.2434 

1.92094 

6.07454 

1.54529 

3.32922 

7.17258 

.271003 

3.70 

13.6900 

50.6530 

1.92354 

6.08276 

1.54668 

3.33222 

7.17905 

.270270 

3.71 

13.7641 

51.0648 

1.92614 

6.09098 

1.54807 

3.33522 

7.18552 

.269542 

3.72 

13.8384 

51.4788 

1.92873 

6.09918 

1.54946 

3.33822 

7.19197 

.268817 

3.73 

13.9129 

51.8951 

1.93132 

6.10737 

1.55085 

3.34120 

7.19841 

.268097 

3.74 

13.9876 

52.3136 

1.93391 

6.11555 

1.55223 

3.34419 

7.20483 

.267380 

3.75 

14.0625 

52.7344 

1.93649 

6.12372 

1.55362 

3.34716 

7.21125 

.266667 

3.76 

14.1376 

53.1574 

1.93907 

6.13188 

1.55500 

3.35014 

7.21765 

.265957 

3.77 

14.2129 

53.5826 

1.94165 

6.14003 

1.55637 

3.35310 

7.22405' 

.265252 

3.78 

14.2884 

54.0102 

1.94422 

6.14817 

1.55775 

3.35607 

7.23043 

.264550 

3.79 

14.3641 

54.4399 

1.94679 

6.15630 

1.55912 

3.35902 

7.23680 

.263852 

3.80 

14.4400 

54.8720 

1.94936 

6.16441 

1.56049 

3.36198 

7.24316 

.263158 

3.81 

14.5161 

55.3063 

1.95192 

6.17252 

1.56186 

3.36492 

7.24950 

.262467 

3.82 

14.5924 

55.7430 

1.95448 

6.18061 

1.56322 

3.36786 

7.25584 

.261780 

3.83 

14.6689 

56.1819 

1.95704 

6.18870 

1.56459 

3.37080 

7.26217 

.261097 

3.84 

14.7456 

56.6231 

1.95959 

6.19677 

1.56595 

3.37373 

7.26848 

.260417 

3.85 

14.8225 

57.0666 

1.96214 

6.20484 

1.56731 

3.37666 

7.27479 

.259740 

3.86 

14.8996 

57.5125 

1.96469 

6.21289 

1.56866 

3.37958 

7.28108 

.259067 

3.87 

14.9769 

57.9606 

1.96723 

6.22093 

1.57001 

3.38249 

7.28736 

.258398 

3.88 

15.0544 

58.4111 

1.96977 

6.22896 

1.57137 

3.38540 

7.29363 

.257732 

3.89 

15.1321 

58.8639 

1.97231 

6.23699 

1.57271 

3.38831 

7.29989 

.257069 

3.90 

15.2100 

59.3190 

1.97484 

6.24500 

1.57406 

3.39121 

7.30614 

.256410 

3.91 

15.2881 

59.7765 

1.97737 

6.25300 

1.57541 

3.39411 

7.31238 

.255755 

3.92 

15.3664 

60.2363 

1.97990 

6.26099 

1.57675 

3.39700 

7.31861 

.255102 

3.93 

15.4449 

60.6985 

1.98242 

6.26897 

1.57809 

3.39988 

7.32483 

.254453 

3.94 

15.5236 

61.1630 

1.98494 

6.27694 

1.57942 

3.40277 

7.33104 

.253807 

3.95 

15.6025 

61.6299 

1 .98746 

6.28490 

1.58076 

3.40564 

7.33723 

.253165 

3.96 

15.6816 

62.0991 

1.98997 

6.29285 

1.58209 

3.40851 

7.34342 

.252525 

3.97 

15.7609 

62.5708 

1.99249 

6.30079 

1.58342 

3.41138 

7.34960 

.251889 

3.98 

15.8404 

63.0448 

1 99499 

6.30872 

1.58475 

3.41424 

7.35576 

.251256 

3.99 

15.9201 

63.5212 

1.99750 

6.31664 

1.58608 

3.41710 

7.36192 

.250627 

4.00 

16.0000 

64.0000 

2.00000 

6.32456 

1.58740 

3.41995 

7.36806 

.250000 

112 d POWERS,  ROOTS,  AND  RECIPROCALS. 


n 

W2 

n3 

A/lO  n 

*!n 

| 'V'lO  n 

j^lOOn 

1 

n 

4.01 

16.0801 

64.4812 

2.00250 

6.33246 

1.58872 

3.42280 

7.37420 

.249377 

4.02 

16.1604 

64.9648 

2.00499 

6.34035 

1.59004 

3.42564 

7.38032 

.248756 

4.03 

16.2409 

65.4508 

2.00749 

6.34823 

1.59136 

3.42848 

7.38644 

.248139 

4.04 

16.3216 

65.9393 

2.00998 

6.35610 

1.59267 

3.43131 

7.39254 

.247525 

4.05 

16.4025 

66.4301 

2.01246 

6.36396 

1.59399 

3.43414 

7.39864 

.246914 

4.06 

16.4836 

66.9234 

2.01494 

6.37181 

1.59530 

3.43697 

7.40472 

.246305 

4.07 

16.5649 

67.4191 

2.01742 

6.37966 

1.59661 

3.43979 

7.41080 

.245700 

4.08 

16.6464 

67.9173 

2.01990 

6.38749 

1.59791 

3.44260 

7.41686 

.245098 

4.09 

16.7281 

68.4179 

2.02237 

6.39531 

1.59922 

3.44541 

7.42291 

.244499 

4.10 

16.8100 

68.9210 

2.02485 

6.40312 

1.60052 

3.44822 

7.42896 

.243902 

4.11 

16.8921 

69.4265 

2.02731 

6.41093 

1.60182 

3.45102 

7.43499 

.243309 

4.12 

16.9744 

69.9345 

2.02978 

6.41872 

1.60312 

3.45382 

7.44102 

.242718 

4.13 

17.0569 

70.4450 

2.03224 

6.42651 

1.60441 

3.45661 

7.44703 

.242131 

4.14 

17.1396 

70.9579 

2.03470 

6.43428 

1.60571 

3.45939 

7.45304 

.241546 

4.15 

17.2225 

71.4734 

2.03715 

6.44205 

1.60700 

3.46218 

7.45904 

.240964 

4.16 

17.3056 

71.9913 

2.03961 

6.44981 

1.60829 

3.46496 

7.46502 

.240385 

4.17 

17.3889 

72.5117 

2.04206 

6.45755 

1.60958 

3.46773 

7.47100 

.239808 

4.18 

17.4724 

73.0346 

2.04450 

6.46529 

1.61086 

3.47050 

7.47697 

.239234 

4.19 

17.5561 

73.5601 

2.04695 

6.47302 

1.61215 

3.47327 

7.48292 

.238664 

4.20 

17.6400 

74.0880 

2.04939 

6.48074 

1.61343 

3.47603 

7.48887 

.238095 

4.21 

17.7241 

74.6185 

2.05183 

6.48845 

1.61471 

3.47878 

7.49481 

.237530 

4.22 

17.8084 

75.1514 

2.05426 

6.49615 

1.61599 

3.48154 

7.50074 

.236967 

4.23 

17.8929 

75.6870 

2.05670 

6.50385 

1.61726 

3.48428 

7.50666 

.236407 

4.24 

17.9776 

76.2250 

2.05913 

6.51153 

1.61853 

3.48703 

7.51257 

.235849 

4.25 

18.0625 

76.7656 

2.06155 

6.51920 

1.61981 

3.48977 

7.51847 

.235294 

4.26 

18.1476 

77.3088 

2.06398 

6.52687 

1.62108 

3.49250 

7.52437 

.234742 

4.27 

18.2329 

77.8545 

2.06640 

6.53452 

1.62234 

3.49523 

7.53025 

.234192 

4.28 

18.3184 

78.4028 

2.06882 

6.54217 

1.62361 

3.49796 

7.53612 

.233645 

4.29 

18.4041 

78.9536 

2.07123 

6.54981 

1.62487 

3.50068 

7.54199 

.233100 

4.30 

18.4900 

79.5070 

2.07364 

6.55744 

1.62613 

3.50340 

7.54784 

.232558 

4.31 

18.5761 

80.0630 

2.07605 

6.56506 

1.62739 

3.50611 

7.55369 

.232019 

4.32 

18.6624 

80.6216 

2.07846 

6.57267 

1.62865 

3.50882 

7.55953 

.231482 

4.33 

18.7489 

81.1827 

2.08087 

6.58027 

1.62991 

3.51153 

7.56535 

.230947 

4.34 

18.8356 

81.7465 

2.08327 

6.58787 

1.63116 

3.51423 

7.57117 

.230415 

4.35 

18.9225 

82.3129 

2.08567 

6.59545 

1.63241 

3.51692 

7.57698 

.229885 

4.36 

19.0096 

82.8819 

2.08806 

6.60303 

1.63366 

3.51962 

7.58279 

.229358 

4.37 

19.0969 

83.4535 

2.09045 

6.61060 

1.63491 

3.52231 

7.58858 

.228833 

4.38 

19.1844 

84.0277 

2.09284 

6.61816 

1.63616 

3.52499 

7.59436 

.228311 

4.39 

19.2721 

84.6045 

2.09523 

6.62571 

1.63740 

3.52767 

7.60014 

.227790 

4.40 

19.3600 

85.1840 

2.09762 

6.63325 

1.63864 

3.53035 

7.60590 

.227273 

4.41 

19.4481 

85.7661 

2.10000 

6.64078 

1.63988 

3.53302 

7.61166 

.226757 

4.42 

19.5364 

86.3509 

2.10238 

6.64831 

1.64112 

3.53569 

7.61741 

.226244 

4.43 

19.6249 

86.9383 

2.10476 

6.65582 

1.64236 

3.53835 

7.62315 

.225734 

4.44 

19.7136 

87.5284 

2.10713 

6.66333 

1.64359 

3.54101 

7.62888 

.225225 

4.45 

19.8025 

88.1211 

2.10950 

6.67083 

1.64483 

3.54367 

7.63461 

.224719 

4.46 

19.8916 

88.7165 

2.11187 

6.67832 

1.64606 

3.54632 

7.64032 

.224215 

4.47 

19.9809 

89.3146 

2.11424 

6.68581 

1.64729 

3.54897 

7.64603 

.223714 

4.48 

20.0704 

89.9154 

2.11660 

6.69328 

1.64851 

3.55162 

7.65172 

.223214 

4.49 

20.1601 

90.5188 

2.11896 

6.70075 

1.64974 

3.55426 

7.65741 

.222717 

4.50 

20.2500 

91.1250 

2.12132 

6.70820 

1.65096 

3.55689 

7.66309 

.222222 

POWERS,  ROOTS,  AND  RECIPROCALS.  U2e 


1 

n 

71* 

n3 

$10  n 

i 'In 

\To7i 

$10071 

n 

4.51 

20.3401 

91.7339 

2.12368 

6.71565 

1.65219 

3.55953 

7.66877 

.221730 

4.52 

20.4304 

92.3454 

2.12603 

6.72309 

1.65341 

3.56215 

7.67443 

.221239 

4.53 

20.5209 

92.9597 

2.12838 

6.73053 

1.65462 

3.56478 

7.68009 

.220751 

4.54 

20.6116 

93.5767 

2.13073 

6.73795 

1.65584 

3.56740 

7.68573 

.220264 

4.55 

20.7025 

94.1964 

2.13307 

6.74537 

1.65706 

3.57002 

7.69137 

.219780 

4.56 

20.7936 

94.8188 

2.13542 

6.75278 

1.65827 

3.57263 

7.69700 

.219298 

4.57 

20.8849 

95.4440 

2.13776 

6.76018 

1.65948 

3.57524 

7.70262 

.218818 

4.58 

20.9764 

96.0719 

2.14009 

6.76757 

1.66069 

3.57785 

7.70824 

.218341 

4.59 

21.0681 

96.7026 

2.14243 

6.77495 

1.66190 

3.58045 

7.71384 

.217865 

4.60 

21.1600 

87.3360 

2.14476 

6.78233 

1.66310 

3.58305 

7.71944 

.217391 

4.61 

21.2521 

97.9722 

2.14709 

6.78970 

1.66431 

3.58564 

7.72503 

.216920 

4.62 

21.3444 

98.6111 

2.14942 

6.79706 

1.66551 

3.58823 

7.73061 

.216450 

4.63 

21.4369 

99.2528 

2.15174 

6.80441 

1.66671 

3.59082 

7.73619 

.215983 

4.64 

21.5296 

99.8973 

2.15407 

6.81175 

1.66791 

3.59340 

7.74175 

.215517 

4.65 

21.6225 

100.545 

2.15639 

6.81909 

1.66911 

3.59598 

7.74731 

.215054 

4.66 

21.7156 

101.195 

2.15870 

6.82642 

1.67030 

3.59856 

7.75286 

.214592 

4.67 

21.8089 

101.848 

2.16102 

6.83374 

1.67150 

3.60113 

7.75840 

.214133 

4.68 

21.9024 

102.503 

2.16333 

6.84105 

1.67269 

3.60370 

7.76394 

.213675 

4.69 

21.9961 

103.162 

2.16564 

6.84836 

1.67388 

3.60626 

7.76946 

.213220 

4.70 

22.0900 

103.823 

2.16795 

6.85565 

1.67507 

3.60883 

7.77498 

.212766 

4.71 

22.1841 

104.487 

2.17025 

6.86294 

1.67626 

3.61138 

7.78049 

.212314 

4.72 

22.2784 

105.154 

2.17256 

6.87023 

1.67744 

3.61394 

7.78599 

.211864 

4.73 

22.3729 

105.824 

2.17486 

6.87750 

1.67863 

3.61649 

7.79149 

.211417 

4.74 

22.4676 

106.496 

2.17715 

6.88477 

1.67981 

3.61904 

7.79697 

.210971 

4.75 

22.5625 

107.172 

2.17945 

6.89202 

1.68099 

3.62158 

7.80245 

.210526 

4.76 

22.6576 

107.850 

2.18174 

6.89928 

1.68217 

3.62412 

7.80793 

.210084 

4.77 

22.7529 

108.531 

2.18403 

6.90652 

1.68334 

3.62665 

7.81339 

.209644 

4.78 

22.8484 

109.215 

2.18632 

6.91375 

1.68452 

3.62919 

7.81885 

.209205 

4.79 

22.9441 

109.902 

2.18861 

6.92098 

1.68569 

3.63171 

7.82429 

.208768 

4.80 

23.0400 

110.592 

2.19089 

6.92820 

1.68687 

3.63424 

7.82974 

.208333 

4.81 

23.1361 

111.285 

2.19317 

6.93542 

1.68804 

3.63676 

7.83517 

.207900 

4.82 

23.2324 

111.980 

2.19545 

6.94262 

1.68920 

3.63928 

7.84059 

.207469 

4.83 

23.3289 

112.679 

2.19773 

6.94982 

1.69037 

3.64180 

7.84601 

.207039 

4.84 

23.4256 

113.380 

2.20000 

6.95701 

1.69154 

3.64431 

7.85142 

.206612 

4.85 

23.5225 

114.084 

2.20227 

6.96419 

1.69270 

3.64682 

7.85683 

.206186 

4.86 

23.6196 

114.791 

2.20454 

6.97137 

1.69386 

3.64932 

7.86222 

.205761 

4.87 

23.7169 

115.501 

2.20681 

6.97854 

1.69503 

3.65182 

7.86761 

.205339 

4.88 

23.8144 

116.214 

2.20907 

6.98570 

1.69619 

3.65432 

7.87299 

.204918 

4.89 

23.9121 

116.930 

2.21133 

6.99285 

1.69734 

3.65682 

7.87837 

.204499 

4.90 

24.0100 

117.649 

2.21359 

7.00000 

1.69850 

3.65931 

7.88374 

.204082 

4.91 

24.1081 

118.371 

2.21585 

7.00714 

1.69965 

3.66179 

7.88909 

.203666 

4.92 

24.2064 

119.095 

2.21811 

7.01427 

1.70081 

3 66428 

7.89445 

.203252 

4.93 

24.3049 

119.823 

2.22036 

7.02140 

1.70196 

3.66676 

7.89979 

.202840 

4.94 

24.4036 

120.554 

2 22261 

7.02851 

1.70311 

3.66924 

7.90513 

.202429 

4.95 

24.5025 

121.287 

2.22486 

7.03562 

1.70426 

3.67171 

7.91046 

.202020 

4.96 

24.6016 

122.024 

2.22711 

7.04273 

1.70540 

3.67418 

7.91578 

.201613 

4.97 

24.7009 

122.763 

2.22935 

7.04982 

1.70655 

3.67665 

7.92110 

.201207 

4.98 

24.8004 

123.506 

2.23159 

7.05691 

1.70769 

3.67911 

7.92641 

.200803 

4.99 

24.9001 

124.251 

2.23383 

7.06399 

1.70884 

3.68157 

7.93171 

.200401 

5.00 

25.0000 

125.000 

2.23607 

7.07107 

1.70998 

3.68403 

7.93701 

.200000 

112/  POWERS,  ROOTS,  AND  RECIPROCALS. 


1 

n 

n 

n2 

n 3 

■Vio  n 

^10  n 

■v'100  n 

5.01 

25.1001 

125.752 

2.23830 

7.07814 

1.71112 

3.68649 

7.94229 

.199601 

5.02 

25.2004 

126.506 

2.24054 

7.08520 

1.71225 

3.68894 

7.94757 

.199203 

5.03 

25.3009 

127.264 

2.24277 

7.09225 

1.71339 

3.69138 

7.95285 

.198807 

5.04 

25.4016 

128.024 

2.24499 

7.09930 

1.71452 

3.69383 

7.95811 

.198413 

5.05 

25.5025 

128.788 

2.24722 

7.10634 

1.71566 

3.69627 

7.96337 

.198020 

5.06 

25.6036 

129.554 

2.24944 

7.11337 

1.71679 

3.69871 

7.96863 

.197629 

5.07 

25.7049 

130.324 

2.25167 

7.12039 

1.71792 

■3.70114 

7.97387 

.197239 

5.08 

25.8064 

131.097 

2.25389 

7.12741 

1.71905 

3.70358 

7.97911 

.196850 

5.09 

25.9081 

131.872 

2.25610 

7.13442 

1.72017 

3.70600 

7.98434 

.196464 

5.10 

26.0100 

132.651 

2.25832 

7.14143 

1.72130 

3.70843 

7.98957 

.196078 

5.11 

26.1121 

133.433 

2.26053 

7.14843 

1.72242 

3.71085 

7.99479 

.195695 

5.12 

26.2144 

134.218 

2.23274 

7.15542 

1.72355 

3.71327 

8.00000 

.195313 

5.13 

26.3169 

135.006 

2.26495 

7.16240 

1.72467 

3.71566 

8.00520 

.194932 

5.14 

26.4196 

135.797 

2.26716 

7.16938 

1.72579 

3.71810 

8.01040 

.194553 

5.15 

26.5225 

136.591 

2.26936 

7.17635 

1.72691 

3.72051 

8.01559 

.194175 

5.16 

26.6256 

137.388 

2.27156 

7.18331 

1.72802 

3.72292 

8.02078 

.193798 

5.17 

26.7289 

138.188 

2.27376 

7.19027 

1.72914 

3.72532 

8.02596 

.193424 

5.18 

26.8324 

138.992 

2.27596 

7.19722 

1.73025 

3.72772 

8.03113 

.193050 

5.19 

26.9361 

139.798 

2.27816 

7.20417 

1.73137 

3.73012 

8.03629 

.192678 

5.20 

27.0400 

140.608 

2.28035 

7.21110 

1.73248 

3.73251 

8.04145 

.192308 

5.21 

27.1441 

141.421 

2.28254 

7.21803 

1.73359 

3.73490 

8.04660 

.191939 

5.22 

2.7.2484 

142.237 

| 2.28473 

7.22496 

1.73470 

3.73729 

8.05175 

.191571 

5.23 

27.3529 

143.056 

[ 2.28692 

7.23187 

1:73580 

3.73968 

8.05689 

.191205 

5.24 

27.4576 

143.878 

2.28910 

7.23878 

1.23691 

3.74206 

8.06202 

.190840 

5.25 

27.5625 

144.703  . 

j 2.29129 

7.24569 

1.73801 

3.74443 

8.06714 

.190476 

5.26 

27.6676 

145.532 

! 2.29347 

7.25259 

1.73912 

3.74681 

8.07226  ! 

.190114 

5.27 

27.7729 

146.363 

2,29565 

7.25948 

1.74022 

3.74918 

8.07737 

.189753 

5.28 

27.8784 

147.198 

! 2.29783 

7.26636 

1.74132 

3.75158 

8,08248 

.189394 

5.29 

27.9841 

148.036 

2.30000 

7.27324 

1.74242 

3.75392 

8.08758 

.189036 

5.30 

28.0900 

148.877 

2.30217 

7.28011 

1.74351 

3.75629 

8.09267 

.188679 

5.31 

28.1961 

149.721 

2.30434 

7.28697 

1.74461 

3.75865 

8.09776 

.188324 

5.32 

28.3024 

150.569 

2.30651 

7.29383 

1.74570 

3.76100 

8.10284 

.187970 

5.33 

28.4089 

151.419 

2.30868 

7.30068 

1.74680 

3.76336 

8.10791 

.187617- 

5.34 

28.5156 

152.273 

2.31084 

7.30753 

1.74789 

3.76571 

8.11298 

.187266 

5.35 

28.6225 

153.130 

2.31301 

7.31437 

1.74898 

3.76806 

8.11804 

.186916 

5.36 

28.7296 

153.991 

2.31517 

7.32120 

1.75007 

3.77041 

8.12310 

.186567 

5.37 

28.8369 

154.854 

2.31733 

7.32803 

1.75116 

3.77275 

8.12814 

.186220 

5.38 

28.9444 

155.721 

2.31948 

7.33485 

1.75224 

3.77509 

8.13319 

.185874 

5.39 

29.0521 

156.591 

2.32164 

7.34166 

1.75333 

3.77740 

8.13822 

.185529 

5.40 

29.1600 

157.464 

2.32379 

7.34847 

1.75441 

3.77976 

8.14325 

.185185 

5.41 

29.2681 

158.340 

2.32594 

7.35527 

1.75549 

3.78210 

8.14828 

.184843 

5.42 

29.3764 

159.220 

2.32809 

7.36206 

1.75657 

3.78442 

8.15329 

.184502 

5.43 

29.4849 

160.103 

2.33024 

7.36885 

1.75765 

3.78675 

8.15831 

.184162 

5.44 

29.5936 

160.989 

2.33238 

7.37564 

1.75873 

3.78907 

8.16331 

.183824 

5.45 

29.7025 

161.879 

2.33452 

7.38241 

1.75981 

3.79139 

8.16831 

.183486 

5.46 

29.8116 

162.771 

2.33666 

7.38918 

1 .76088 

3.79371 

8.17330 

.183150 

5.47 

29.9209 

163.667 

2.33880 

7.39594 

1.76196 

3.79603 

8.17829 

.182815 

5.48 

30.0304 

164.567 

2.34094 

7.40270 

1.76303 

3.79834 

8.18327 

.182482 

5.49 

30.1401 

165.469 

2.34307 

7.40945 

1.76410 

3.80065 

8.18824 

.182149 

5.50 

30.2500 

166.375 

2.34521 

7.41620 

1.76517 

3.80295 

8.19321 

.181818 

POWERS,  ROOTS,  AND  RECIPROCALS.  112^ 


n 

7*2 

71* 

>/l0  n 

^10  71  1 

! 

\ylmn 

1 

! 71 

5.51 

30.3601 

167.284 

2.34734 

7.42294 

1.76624 

1 

3.80526 

8.19818 

.181488 

5.52 

30.4704 

168.197 

2.34947 

7.42967 

1.76731 

3.80756 

8.20313 

.181159 

5.53 

30.5809 

169.112 

2.35160 

7.43640 

1.76838 

3.80986 

8.20808 

.180832 

5.54 

30.6916 

170.031 

2.35372 

7.44312 

1.76944 

3.80115  | 

8.21303 

.180505 

5.55 

30.8025 

170.954 

2.35584 

7.44983 

1.77051 

3.81444 

8.21797 

.180180 

5.56 

30.9136 

171.880 

2.35797 

7.45654 

1.77157 

3.81673 

8.22290 

.179856 

5.57 

31.0249 

172.809 

2.36008 

7.46324 

1.77263 

3.81902 

8.22783 

.179533 

5.58 

31.1364 

173.741 

2.36220 

7.46994 

1.77369 

3.82130 

8.23275 

.179212 

5.59 

31.2481 

174.677 

2.36432 

7.47663 

1.77475 

3.82358 

| 8.23766 

.178891 

5.60 

31.3600 

175.616 

2.36643 

7.48331 

1.77581 

3.82586 

8.24257 

.178571 

5.61 

31.4721 

176.558 

2.36854 

7.48999 

1.77686 

3.82814 

8.24747 

.178253 

5.62 

31.5844 

177.504 

2.37065 

7.49667 

1.77792 

3.83041 

8.25237 

.177936 

5.63 

31.6969 

178.454 

2.37276 

7.50333 

1.77897 

3.83268 

8.25726 

.177620 

5.64 

31.8096 

179.406 

2.37487 

7.50999 

1.78003 

3.83495 

8.26215 

.177305 

5.65 

31.9225 

180.362 

2.37697 

7.51665 

1.78108 

3.83721 

8.26703 

.176991 

5.66 

32.0356 

181.321 

2.37908 

7.52330 

1.78213 

3.83948 

8.27190 

.176678 

5.67 

32.1489 

182.284 

2.38118 

7.52994 

1.78318 

3.84174 

8.27677 

.176367 

5.68 

32.2624 

183.250 

2.38328 

7.53658 

1.78422 

3.84400 

8.28164 

.176056 

5.69 

32.3761 

184.220 

2.38537 

7.54321 

1.78527 

3.84625 

8.28649 

.175747 

5.70 

32.4900 

185.193 

2.38747 

7.54983 

1.78632 

3.84850 

8.29134 

.175439 

5.71 

32.6041 

186.169 

2.38956 

7.55645 

1.78736 

3.85075 

8.29619 

.175131 

5.72 

32.7184 

187.149 

2^39165 

7.56307 

1.78840 

3.85300 

8.30103 

.174825 

5.73 

32.8329 

188.133 

2.39374 

7.56968 

1.78944 

3.85524 

8.30587 

.174520 

5.74 

32.9476 

189.119 

2.39583 

7.57628 

1.79048 

3.85748 

8.31069 

.174216 

5.75 

33.0625 

190.109 

2.39792 

7.58288 

1.79152 

3.85972 

8.31552 

.173913 

5.76 

33.1776 

191.103 

2.40000 

7.58947 

1.79256 

3.86196 

8.32034 

.173611 

5.77 

33.2929 

192.100 

2.40208 

7.59605 

1.79360 

3.86419 

8.32515 

.173310 

5.78 

33.4084 

193.101 

2.40416 

7.60263 

1.79463 

3.86642 

8.32995 

.173010 

5.79 

33.5241 

194.105 

2.40624 

7.60920 

1.79567 

3.86865 

8.33476 

.172712 

5.80 

33.6400 

195.112 

2.40832 

7.61577 

1.79670 

3.87088 

8.33955 

.172414 

5.81 

33.7561 

196.123 

2.41039 

7.62234 

1.79773 

3.87310 

8.34434 

.172117 

5.82 

33.8724 

197.137 

2.41247 

7.62889 

1.79876 

3.87532 

8.34913 

.171821 

5.83 

33.9889 

198.155 

2.41454 

7.63544 

1.79979 

3.87754 

8.35390 

.171527 

5.84 

34.1056 

199.177 

2.41661 

7.64199 

1.80082 

3.87975 

8.35868 

.171233 

5.85 

34.2225 

200.202 

2.41868 

7.64853 

1.80185 

3.88197 

8.36345 

.170940 

5.86 

34.3396 

201.230 

2.42074 

7.65506 

1.80288 

3.88418 

8.36821 

.170649 

5.87 

34.4569 

202.262 

2.42281 

7.66159 

1.80390 

3.88639 

8.37297 

.170358 

5.88 

34.5744 

203.297 

2.42487 

7.66812 

1.80492 

3.88859 

8.37772 

.170068 

5.89 

34.6921 

204.336 

2.42693 

7.67463 

1.80595 

3.89082 

8.38247 

.169779 

5.90 

34.8100 

205.379 

2.42899 

7.68115 

1.80697 

3.89300 

8.38721 

.169492 

5.91 

34.9281 

206.425 

2.43105 

7.68765 

1.80799 

3.89520 

8.39194 

.169205 

5.92 

35.0464 

207.475 

2.43311 

7.69415 

1.80901 

3.89739 

8.39667 

.168919 

5.93 

35.1649 

208.528 

2.43516 

7.70065 

1.81003 

3.89958 

8.40140 

.168634 

5.94 

35.2836 

209.585 

2.43721 

7.70714 

1.81104 

3.90177 

8.40612 

.168350 

5.95 

35.4025 

210.645 

2.43926 

7.71362 

1.81206 

3.90396 

8.41083 

.168067 

5.96 

35.5216 

211.709 

2.44131 

7.72010 

1.81307 

3.90615 

8.41554 

.167785 

5.97 

35.6409 

212.776 

2.44336 

7.72658 

1.81409 

3.90833 

8.42025 

.167504 

5.98 

35.7604 

213.847 

2.44540 

7.73305 

1.81510 

3.91051 

8.42494 

.167224 

5.99 

35.8801' 

214.922 

2.44745 

7.73951 

1.81611 

3.91269 

8.42964 

.166945 

6.00 

36.0000 

216.000 

2.44949 

7.74597 

1.81712 

3.91487 

8.43433 

.166667 

11 2h  POWERS,  ROOTS,  AND  RECIPROCALS. 


n 

tt2 

tt3 

AS 

VTOn 

fllOn 

1 £ 
8 

1 

n 

6.01 

36.1201 

217.082 

2.45153 

7.75242 

1.81813 

3.91704 

8.43901 

.166389 

6.02 

36.2404 

218.167 

2.45357 

7.75887 

1.81914 

3.91921 

8.44369 

.166113 

6.03 

36.3609 

219.256 

2.45561 

7.76531 

1.82014 

3.92138 

8.44836 

.165838 

6.04 

36.4816 

220.349 

2.45764 

7.77174 

1.82115 

3.92355 

8.45303 

.165563 

6.05 

36.6025 

221.445 

2.45967 

7.77817 

1.82215 

3.92571 

8.45769 

.165289 

6.06 

36.7236 

222.545 

2.46171 

7.78460 

1.82316 

3.92787 

8.46235 

.165017 

6.07 

36.8449 

223.649 

2.46374 

7.79102 

1.82416 

3.93003 

8.46700 

.164745 

6.08 

36.9664 

224.756 

2.46577 

7.79744 

1.82516 

3.93219 

8.47165 

.164474 

6.09 

37.0881 

225.867 

2.46779 

7.80385 

1.82616 

3.93434 

8.47629 

.164204 

6.10 

37.2100 

226.981 

2.46982 

7.81025 

1.82716 

3.93650 

8.48093 

.163934 

6.11 

37.3321 

228.099 

2.47184 

7.81665 

1.82816 

3.93865 

8.48556 

.163666 

6.12 

37.4544 

229.221 

2.47386 

7.82304 

1.82915 

3.94079 

8.49018 

.163399 

6.13 

37.5769 

230.346 

2.47588 

7.82943 

1.83015 

3.94294 

8.49481 

.163132 

6.14 

37.6996 

231.476 

2.47790 

7.83582 

1.83115 

3.94508 

8.49942 

.162866 

6.15 

37.8225 

232.608 

2.47992 

7.84219 

1.83214 

3.94722 

8.50404 

.162602 

6.16 

37.9456 

233.745 

2.48193 

7.84857 

1.83313 

3.94936 

8.50864 

.162338 

6.17 

38.0689 

234.885 

2.48395 

7.85493 

1.83412 

3.95150 

8.51324 

.162075 

6.18 

38.1924 

236.029 

2.48596 

7.86130 

1.83511 

3.95363 

8.51784 

.161812 

6.19 

38.3161 

237.177 

2.48797 

7.86766 

1.83610 

3.95576 

8.52243 

.161551 

6.20 

38.4400 

238.328 

2.48998 

7.87401 

1.83709 

3.95789 

8.52702 

.161290 

6.21 

38.5641 

239.483 

2.49199 

7.88036 

1.83808 

3.96002 

8.53160 

.161031 

6.22 

38.6884 

240.642 

2.49399 

7.88670 

1.83906 

3.96214 

8.53618 

.160772 

6.23 

38.8129 

241.804 

2.49600 

7.89303 

1.84005 

3.96426 

8.54075 

.160514 

6.24 

38.9376 

242.971 

2.49800 

7.89937 

1.84103 

3.96639 

8.54532 

.160256 

6.25 

39.0625 

244.141 

2.50000 

7.90569 

1.84202 

3.96850 

8.54988 

.160000 

6.26 

39.1876 

245.314 

2.50200 

7.91202 

1.84300 

3.97062 

8.55444 

.159744 

6.27 

39.3129 

246.492 

2.50400 

7.91833 

1.84398 

3.97273 

8.55899 

.159490 

6.28 

39.4384 

247.673 

2.50599 

7.92465 

1.84496 

3.97484 

8.56354 

.159236 

6.29 

39.5641 

248.858 

2.50799 

7.93095 

1.84594 

3.97695 

8.56808 

.158983 

6.30 

39.6900 

250.047 

2.50998 

7.93725 

1.84691 

3.97906 

8.57262 

.158730 

6.31 

39.8161 

251.240 

2.51197 

7.94355 

1.84789 

3.98116 

8.57715 

.158479 

6.32 

39.9424 

252.436 

2.51396 

7.94984 

1.84887 

3.98326 

8.58168 

.158228 

6.33 

40.0689 

253.636 

2.51595 

7.95613 

1.84984 

3.98536 

8.58620 

.157978 

6.34 

40.1956 

254.840 

2.51794 

7.96241 

1.85082 

3.98746 

8.59072 

.157729 

6.35 

40.3225 

256.048 

2.51992 

7.96869 

1.85179 

3.98956 

8.59524 

.157480 

6.36 

40.4496 

257.259 

2.52190 

7.97496 

1.85276 

3.99165 

8.59975 

.157233 

6.37 

40.5769 

258.475 

2.52389 

7.98123 

1.85373 

3 99374 

8.60425 

.156986 

6.38 

40.7044 

259.694 

2.52587 

7.98749 

1.85470 

3.99583 

8.60875 

.156740 

6.39 

40.8321 

260.917 

2.52784 

7.99375 

1.85567 

3.99792 

8.61325 

.156495 

6.40 

40.9600 

262.144 

2.52982 

8.00000 

1.85664 

4.00000 

8.61774 

.156250 

6.41 

41.0881 

263.375 

2.53180 

8.00625 

1.85760 

4.00208 

8.62222 

.156006 

6.42 

41.2164 

264.609 

2.53377 

8.01249 

1.85857 

4.00416 

8.62671 

.155763 

6.43 

41.3449 

265.848 

2.53574 

8.01873 

1.85953 

4.00624 

8.63118 

.155521 

6.44 

41.4736 

267.090 

2.53772 

8.02496 

1.86050 

4.00832 

8.63566 

.155280 

6.45 

41.6025 

268.336 

2.53969 

8.03119 

1.86146 

4.01039 

8.64012 

.155039 

6.46 

41.7316 

269.586 

2.54165 

8.03741 

1.86242 

4.01246 

8.64459 

.154799 

6.47 

41.8609 

270.840 

2.54362 

8.04363 

1.86338 

4.01453 

8.64904 

.154560 

6.48 

41.9904 

272.098 

2.54558 

8.04984 

1.86434 

4.01660 

8.65350 

.154321 

6.49 

42.1201 

273.359 

2.54755 

8.05605 

1.86530 

4.01866 

8.65795 

.154083 

6.50 

42.2500 

274.625 

2.54951 

8.06226 

1.86626 

4.02073 

8.66239 

.153846 

POWERS,  ROOTS,  AND  RECIPROCALS.  112l 


Wn 

1 

n 

n3 

V/l 

VlO  n 

^100  n 

n 

6.51 

42.3801 

275.894 

2.55147 

8.06846 

1.86721 

4.02279 

8.66683 

.153610 

6.52 

42.5104 

277.168 

2.55343 

8.07465 

1.86817 

4.02485 

8.67127 

.153374 

6.53 

42.6409 

278.445 

2.55539 

8.08084 

1.86912 

4.02690 

8.67570 

.153139 

6.54 

42.7716 

279.726 

2.55734 

8.08703 

1.87008 

4.02896 

8.68012 

.152905 

6.55 

42.9025 

281.011 

2.55930 

8.09321 

1.87103 

4.03101 

8.68455 

.152672 

6.56 

43.0336 

282.300 

2.56125 

8.09938 

1.87198 

4.03306 

8.68896 

.152439 

6.57 

43.1649 

283.593 

2.56320 

8.10555 

1.87293 

4.03511 

8.69338 

.152207 

6.58 

43.2964 

284.890 

2.56515 

8.11172 

1.87388 

4.03715 

8.69778 

.151976 

6.59 

43.4281 

286.191 

2.56710 

8.11788 

1.87483 

4.03920 

8.70219 

.151745 

6.60 

43.5600 

287.496 

2.56905 

8.12404 

1.87578 

4.04124 

8.70659 

.151515 

6.61 

43.6921 

288.805 

2.57099 

8.13019 

1.87672 

4.04328 

8.71098 

.151286 

6.62 

43.8244 

290.118 

2.57294 

8.13634 

1.87767 

4.04532 

8.71537 

.151057 

6.63 

43.9569 

291.434 

2.57488 

8.14248 

1.87862 

4.04735 

8.71976 

.150830 

6.64 

44.0S96 

292.755 

2.57682 

8.14862 

1.87956 

4.04939 

8.72414 

.150602 

6.65 

44.2225 

294.080 

2.57876 

8.15475 

1.88050 

4.05142 

8.72852 

.150376 

6.66 

44.3556 

295.408 

2.58070 

8.16088 

1.88144 

4.05345 

8.73289 

.150150 

6.67 

44.48S9 

296.741 

2.58263 

8.16701 

1.88239 

4.05548 

8.73726 

.149925 

6.68 

44.6224 

298.078 

2.58457 

8.17313 

1.88333 

4.05750 

8.74162 

.149701 

6.69 

44.7561 

299.418 

2.58650 

8.17924 

1.88427 

4.05953 

8.74598 

.149477 

6.70 

44.8900 

300.763 

2.58844 

8.18535 

1.88520 

4.06155 

8.75034 

.149254 

6.71 

45.0241 

302.112 

2.59037 

8.19146 

1.88614 

4.06357 

8.75469 

.149031 

6.72 

45.1584 

303.464 

2.59230 

8.19756 

1.88708 

4.06558 

8.75904 

.148810 

6.73 

45.2929 

304.821 

2.59422 

8.20366 

1.88801 

4.06760 

8.76338 

.148588 

6.74 

45.4276 

306.182 

2.59615 

8.20975 

1.88895 

4.06961 

8.76772 

.148368 

6.75 

45.5625 

307.547 

2.59808 

8.21584 

1.88988 

4.07163 

8.77205 

.148148 

6.76 

45.6976 

308.916 

2.60000 

8.22192 

1.89081 

4.07364 

8.77638 

.147929 

6.77 

45.8329 

310.289 

2.60192 

8.22800 

1.89175 

4.07564 

8.78071 

.147711 

6.78 

45.9684 

311.666 

2.60384 

8.23408 

1.89268 

4.07765 

8.78503 

.147493 

6.79 

46.1041 

313.047 

2.60576 

8.24015 

1.89361 

4.07965 

8.78935 

.147275 

6.80 

46.2400 

314.432 

2.60768 

8.24621 

1.89454 

4.08166 

8.79366 

.147059 

6.81 

46.3761 

315.821 

2.60960 

8.25227 

1.89546 

4.08365 

8.79797 

.146843 

6.82 

46.5124 

! 317.215 

2.61151 

8.25833 

1.89639 

4.08565 

8.80227 

.146628 

6.83 

46.6489 

318.612 

2.61343 

8.26438 

1.89732 

4.08765 

8.80657 

.146413 

6.84 

46.7856 

320.014 

2.61534 

8.27043 

1 .89824 

4.08964 

8.81087 

.146199 

6.85 

46.9225 

321.419 

2.61725 

8.27647 

1.89917 

4.09164 

8.81516 

.145985 

6.86 

47.0596 

322.829 

2.61916 

8.28251 

1.90009 

4.09362 

8.81945 

.145773 

6.87 

47.1969 

324.243 

2.62107 

8.28855 

1.90102 

4.09561 

8.82373 

.145560 

6.88 

47.3344 

325.661 

2.62298 

8.29458 

1.90194 

4.09760 

8.82801 

.145349 

6.89 

47.4721 

327.083 

2.62488 

8.30060 

1.90286 

4.09958 

8.83229 

.145138 

6.90 

47.6100 

328.509 

2.62679 

8.30662 

1.90378 

4.10157 

8.83656 

.144928 

6.91 

47.7481 

329.939 

2.62869 

8.31264 

1.90470 

4.10355 

8.84082 

.144718 

6.92 

47.8864 

331.374 

2.63059 

8.31865 

1.90562 

4.10552 

8.84509 

.144509 

6.93 

48.0249 

332.813 

2.63249 

8.32466 

1.90653 

4.10750 

8.84.934 

.144300 

6.94 

48.1636 

334.255 

2.63439 

8.33067 

1.90745 

4.10948 

8.85360 

.144092 

6.95 

48.3025 

335.702 

2.63629 

8.33667 

1.90837 

4.11145 

8.85785 

.143885 

6.96 

48.4416 

337.154 

2.63818 

8.34266 

1.90928 

4.11342 

8.86210 

.143678 

6.97 

48.5809 

338.609 

2.64008 

8.34865 

1.91019 

4.11539 

8.86634 

.143472 

6.98 

; 48.7204 

340.068 

2.64197 

8.35464 

1.91111 

4.11736 

8.87058 

.143267 

6.99 

48.8601 

341.532 

2.64386 

8.36062 

1.91202 

4.11932 

8.87481 

.143062 

7.00 

, 49.0000 

343.000 

2.64575 

8.36660 

1.91293  | 

4.12129 

8.87904 

.142857 

112;  POWERS,  ROOTS,  AND  RECIPROCALS. 


n 

W2 

W3 

yfn 

*10  n 

7.01 

49.1401 

344.472 

2.64764 

8.37257 

7.02 

49.2804 

345.948 

2.64953 

8.37854 

7.03 

49.4209 

347.429 

2.65141 

8.38451 

7.04 

49.5616 

348.914 

2.65330 

8.39047 

7.05 

49.7025 

350.403 

2.65518 

8.39643 

7.06 

49.8436 

351.896 

2.65707 

8.40238 

7.07 

49.9849 

353.393 

2.65895 

8.40833 

7.08 

50.1264 

354.895 

2.66083 

8.41427 

7.09 

50.2681 

356.401 

2.66271 

8.42021 

7.10 

50.4100 

357.911 

2.66458 

8.42615 

7.11 

50.5521 

359.425 

2.66646 

8.43208 

7.12 

50.6944 

360.944 

2.66833 

8.43801 

7.13 

50.8369 

362.467 

2.67021 

8.44393 

7.14 

50.9796 

363.994 

2.67208 

8.44985 

7.15 

51.1225 

365.526 

2.67395 

8.45577 

7.16 

51.2656 

367.062 

2.67582 

8.46168 

7.17 

51.4089 

368.602 

2.67769 

8.46759 

7.18 

51.5524 

370.146 

2.67955 

8.47349 

7.19 

51.6961 

371.695 

2.68142 

8.47939 

7.20 

51.8400 

373.248 

2.68328 

8.48528 

7.21 

51.9841 

374.805 

2.68514 

8.49117 

7.22 

52.1284 

376.367 

2.68701 

8.49706 

7 23 

52.2729 

377.933 

2.68887 

8.50294 

7.24 

52.4176 

379.503 

2.69072 

8.50882 

7.25 

52.5625 

381-.078 

2.69258 

8.51469 

7.26 

52.7076 

382.657 

2.69444 

8.52056 

7.27 

52.8529 

384.241 

2.69629 

8.52643 

7.28 

52.9984 

385.828 

2.69815 

8.53229 

7.29 

53.1441 

387.420 

2.70000 

8.53815 

7.30 

53.2900 

389.017 

2.70185 

8.54400 

7.31 

53.4361 

390.4>18 

2.70370 

8.54985 

7.32 

53.5824 

392.223 

2.70555 

8.55570 

7.33 

53.7289 

393.833 

2.70740 

8.56154 

7.34 

53.8756 

395.447 

2.70924 

8.56738 

7.35 

54.0225 

397.065 

2.71109 

8.57321 

7.36 

54.1696 

398.688 

2.71293 

8.57904 

7.37 

54.3169 

400.316 

2.71477 

8.58487 

7.38 

54.4644 

401 .947 

2.71662 

8.59069 

7.39 

54.6121 

403.583 

2.71846 

8.59651 

7.40 

54.7600 

405.224 

2.72029 

8.60233 

7.41 

54.9081 

406.869 

2.72213 

8.60814 

7.42 

55.0564 

408.518 

2.72397 

8.61394 

7.43 

55.2049 

410.172 

2.72580 

8.61974 

7.44 

55.3536 

411.831 

2.72764 

8.62554 

7.45 

55.5025 

413.494 

2.72947 

8.63134 

7.46 

55.6516 

415.161 

2.73130 

8.63713 

7.47 

55.8009 

416.833 

2.73313 

8.64292 

7.48 

55.9504 

418.509 

2.73496 

8.64870 

7.49 

56.1001 

420.190 

2.73679 

8.65448 

7.50  1 

56.2500 

421.875 

2.73861 

8.66025 

1.91384 

1.91475 

1.91566 

1.91657 

1.91747 

1.91838 

1.91929 

1.92019 

1.92109 

1.92200 

1.92290 
1.92380 
1.92470 
1 .92560 
1.92650 

1.92740 

1.92829 

1.92919 

1.93008 

1.93098 

1.93187 

1.93277 

1.93366 

1.93455 

1.93544 

1.93633 

1.93722 

1.93810 

1.93899 

1.93988 

1.94076 

1.94165 

1.94253 

1.94341 

1.94430 

1.94518 

1.94606 

1.94694 

1.94782 

1.94870 

1.94957 

1.95045 

1.95132 

1.95220 

1.95307 

1.95395 
1 .95482 
1.95569 
1.95656 
1.95743 


*10  n 


4.12325 

4.12521 

4.12716 

4.12912 

4.13107 

4.13303 

4.13498 

4.13695 

4.13887 

4.14082 

4.14276 

4.14470 

4.14664 

4.14858 

4.15051 

4.15245 

4.15438 

4.15631 

4.15824 

4.16017 

4.16209 

4.16402 

4.16594 

4.16786 

4.16978 

4.17169 

4.17361 

4.17552 

4.17743 

4.17934 

4.18125 

4.18315 

4.18506 

4.18696 

4.18886 

4.19076 

4.19266 

4.19455 

4.19644 

4.19834 

4.20023 

4.20212 

4.20400 

4.20589 

4.20777 

4.20965 

4.21153 

4.21341 

4.21529 

4.21716 


*10071 


8.88327 

8.88749 

8.89171 

8.89592 

8.90013 

8.90434 

8.90854 

8.91274 

8.91693 

8.92112 

8.92531 

8.92949 

8.93367 

8.93784 

8.94201 

8.94618 

8.95034 

8.95450 

8.95866 

8.96281 

8.96696 

8.97110 

8.97524 

8.97938 

8.98351 

8.98764 

8.99176 

8.99588 

9.00000 

9.00411 

9.00822 

9.01233 

9.01643 

9.02053 

9.02462 

9.02871 

9.03280 

9.03689 

9.04097 

9.04504 

9.04911 

9.05318 

9.05725 

9.06131 

9.06537 

9.06942 

9.07347 

9.07752 

9.08156 

9.08560 


1 

n 


.142653 

.142450 

.142248 

.142046 

.141844 

.141643 

.141443 

.141243 

.141044 

.140845 

.140647 

.140449 

.140253 

.140056 

.139860 

.139665 

.139470 

.139276 

.139082 

.138889 

.138696 

.138504 

.138313 

.138122 

.137931 

.137741 

.137552 

.137363 

.137174 

.136986 

.136799 

.136612 

.136426 

.136240 

.136054 

.135870 

.135685 

.135501 

.135318 

.135135 

.134953 

.134771 

.134590 

.134409 

.134228 

.134048 

.133869 

.133690 

.133511 

.133333 


POWERS,  ROOTS,  AND  RECIPROCALS.  lV2k 


n 

n2 

7l3 

VlOn 

ftbOn 

1 

^10  n 

n 

7.51 

56.4001 

423.565 

2.74044 

8.66603 

1.95830 

4.21904 

9.08964 

.133156 

7.52 

56.5504 

425.259 

2.74226 

8.67179 

1.95917 

4.22091 

9.09367 

.132979 

7.53 

56.7009 

426.958 

2.74408 

8.67756 

1.96004 

4.22278 

9.09770 

.132802 

7.54 

56.8516 

428.661 

2.74591 

8.68332 

1.96091 

4.22465 

9.10173 

.132626 

7.55 

57.0025 

430.369 

2.74773 

8.68907 

1.96177 

4.22651 

9.10575 

.132450 

7.56 

57.1536 

432.081 

2.74955 

8.69483 

1.96264 

4.22838 

9.10977 

.132275 

7.57 

57.3049 

433.798 

2.75136 

8.70057 

1.96350 

4.23024 

9.11378 

.132100 

7.58 

57.4564 

435.520 

2.75318 

8.70632 

1.96437 

4.23210 

9.11779 

.131926 

7.59 

57.6081 

437.245 

2.75500 

8.71206 

1.96523 

4.23396 

9.12180 

.131752 

7.60 

57.7600 

438.976 

2.75681 

8.71780 

1.96610 

4.23582 

9.12581 

.131579 

7.61 

57.9121 

440.711 

2.75862 

8.72353 

1.96696 

4.23768 

9.12981 

.131406 

7.62 

58.0644 

442.451 

2.76043 

8.72926 

1.96782 

4.23954 

9.13380 

.131234 

7.63 

58.2169 

444.195 

2.76225 

8.73499 

1.96868 

4.24139 

9.13780 

.131062 

7.64 

58.3696 

445.994 

2.76405 

8.74071 

1.96954 

4.24324 

9.14179 

.130890 

7.65 

58.5225 

447.697 

2.76586 

8.74643 

1.97040 

4.24509 

9.14577 

.130719 

7.66 

58.6756 

449.455 

2.76767 

8.75214 

1.97126 

4.24694 

9.14976 

.130548 

7.67 

58.8289 

451.218 

2.76948 

8.75785 

1.97211 

4.24879 

9.15374 

.130378 

7.68 

58.9824 

452.985 

2.77128 

8.76356 

1.97297 

4.25063 

9.15771 

.130208 

7.69 

59.1361 

454.757 

2.77308 

8.76926 

1.97383 

4.25248 

9.16169 

.130039 

7.70 

59.2900 

456.533 

2.77489 

8.77496 

1.97468 

4.25432 

9.16566 

.129870 

7.71 

59.4441 

458.314 

2.77669 

8.78066 

1.97554 

4.25616 

9.16962 

.129702 

7.72 

59.5984 

460.100 

2.77849 

8.78635 

1.97639 

4.25800 

9.17359 

.129534 

7.73 

59.7529 

461.890 

2.78029 

8.79204 

1.97724 

4.25984 

9.17754 

.129366 

7.74 

59.9076 

463.685 

2.78209 

8.79773 

1.97809 

4.26168 

9.18150 

.129199 

7.75 

60.0625 

465.484 

2.78388 

8.80341 

1.97895 

4.26351' 

9.18545 

.129032 

7.76 

60.2176 

467.289 

2.78568 

8.80909 

1.97980 

4.26534 

9.18940 

.128866 

7.77 

60.3729 

469.097 

2.78747 

8.81476 

1.98065 

4.26717 

9.19335 

.128700 

7.78 

60.5284 

470.911 

2.78927 

8.82043 

1.98150 

4.26900 

9.19729 

.128535 

7.79 

60.6841 

472.729 

2.79106 

8.82610 

1.98234 

4.27083 

9.20123 

.128370 

7.80 

60.8400 

474.552 

2.79285 

8.83176 

1.98319 

4.27266 

9.20516 

.128205 

7.81 

60.9961 

476.380 

2.79464 

8.83742 

1.98404 

4.27448 

9.20910 

.128041 

7.82 

61.1524 

478.212 

2.79643 

8.84308 

1.98489 

4.27631 

9.21303 

.127877 

7.83 

61.3089 

480.049 

2.79821 

8.84873 

1.98573 

4.27813 

9.21695 

.127714 

7.84 

61.4656 

481.890 

2.80000 

8.85438 

1.98658 

4.27995 

9.22087 

.127551 

7.85 

61.6225 

483.737 

2.80179 

8.86002 

1.98742 

4.28177 

9.22479 

.127389 

7.86 

61.7796 

485.588 

2.80357 

8.86566 

1.98826 

4.28359 

9.22871 

.127227 

7.87 

61.9369 

487.443 

2.80535 

8.87130 

1.98911 

4.28540 

9.23262 

.127065 

7.88 

62.0944 

489.304 

2.80713 

8.87694 

1.98995 

4.28722 

9.23653 

.126904 

7.89 

62.2521 

491.169 

2.80891 

8.88257 

1 .99079 

4.28903 

9.24043 

.126743 

7.90 

62.4100 

493.039 

2.81069 

8.88819 

1.99163 

4.29084 

9.24433 

.126582 

7.91 

62.5681 

494.914 

2.81247 

8.89382 

1.99247 

4.29265 

9.24823 

.126422 

7.92 

62.7264 

496.793 

2.81425 

8.89944 

1 1.99331 

4.29446 

9.25213 

.126263 

7.93 

62.8849 

498.677 

2.81603 

8.90505 

1 1.99415 

4.29627 

9.25602 

.126103 

7.94 

63.0436 

500.566 

2.81780 

8.91067 

1.99499 

4.29807 

9.25991 

.125945 

7.95 

63.2025 

502.460 

2.81957 

8.91628 

1.99582 

4.29987 

9.26380 

.125786 

7.96 

63.3616 

504.358 

2.82135 

8.92188 

1.99666 

4.30168 

9.26768 

.125628 

7.97 

63.5209 

506.262 

2.82312 

8.92749 

1 .99750 

4.30348 

9.27156 

.125471 

7.98 

63.6804 

508.170 

2.82489 

8.93308 

1.99833 

4.30528 

9.27544 

.125313 

7.99 

63.8401 

510.082 

2.82666 

8.93868 

1.99917 

4.30707 

9.27931 

.125156 

8.00 

64.0000 

512.000 

2.82843 

8.94427 

2.00000 

4.30887 

9.28318 

.125000 

112 1 POWERS,  ROOTS,  AND  RECIPROCALS. 


» 

ri 2 

W3 

Vw 

VlOn 

^IlOn 

aToo  n 

1 

n 

8.01 

64.1601 

513.922 

2.83019 

8.94986 

2.00083 

4.31066 

9.28704 

.124844 

8.02 

64.3204 

515.850 

2.83196 

8.95545 

2.00167 

4.31246 

9.29091 

.124688 

8.03 

64.4809 

517.782 

2.83373 

8.96103 

2.00250 

4.31425 

9.29477 

.124533 

8.04 

64.6416 

519.718 

2.83549 

8.96660 

2.00333 

4.31604 

9.29862 

.124378 

8.05 

64.8025 

521.660 

2.83725 

8.97218 

2.00416 

4.31783 

9.30248 

.124224 

8.06 

64.9636 

523.607 

2.83901 

8.97775 

2.00499 

4.31961 

9.30633 

.124070 

8.07 

65.1249 

525.558 

2.84077 

8.98332 

2.00582 

4.32140 

9.31018 

.123916 

8.08 

65.2864 

527.514 

2.84253 

8.98888 

2.00664 

4.32818 

9.31402 

.123762 

8.09 

65.4481 

529.475 

2.84429 

8.99444 

2.00747 

4.32497 

9.31786 

.123609 

8.10 

65.6100 

531.441 

2.84605 

9.00000 

2.00830 

4.32675 

9.32170 

.123457 

8.11 

65.7721 

533.412 

2.84781 

9.00555 

2.00912 

4.32853 

9.32553 

.123305 

8.12 

65.9344 

535.387 

2.84956 

9.01110 

2.00995 

4.33031 

9.32936 

.123153 

8.13 

66.0969 

537.368 

2.85132 

9.01665 

2.01078 

4.33208 

9.33319 

.123001 

8.14 

66.2596 

539.353 

2.85307 

9.02219 

2.01160 

4.33386 

9.33702 

.122850 

8.15 

• 66.4225 

541.343 

2.85482 

9.02774 

2.01242 

4.33563 

9.34084 

.122699 

8.16 

66.5856 

543.338 

2.85657 

9.03327 

2.01325 

4.33741 

9.34466 

.122549 

8.17 

66.7489 

545.339 

2.85832 

9.03881 

2.01407 

4.33918 

9.34847 

.122399 

8.18 

66.9124 

547.343 

2.86007 

9.04434 

2.01489 

4.34095 

9.35229 

.122249 

8.19 

67.0761 

549.353 

2.86182 

9.04986 

2.01571 

4.34272 

9.35610 

.122100 

8.20 

67.2400 

551.368 

2.86356 

9.05539 

2.01653 

4.34448 

9.35990 

.121951 

8.21 

67.4041 

553.388 

2.86531 

9.06091 

2.01735 

4.34625 

9.36370 

.121803 

8.22 

67.5684 

555.412 

2.86705 

9.06642 

2.01817 

4.34801 

9.36751 

.121655 

8.23 

67.7329 

557.442 

2.86880 

9.07193 

2.01899 

4.34977 

9.37130 

.121507 

8.24 

67.8976 

559.476 

2.87054 

9.07744 

2.91980 

4.35153 

9.37510 

.121359 

8.25 

68.0625 

561.516 

2.87228 

9.08295 

2.02062 

4.35329 

9.37889 

.121212 

8.26 

68.2276 

563.560 

2.87402 

9.08845 

2.02144 

4.35505 

9.38268 

.121065 

8.27 

68.3929 

565.609 

2.87576 

9.09395 

2.02225 

4.35681 

9.38646 

.120919 

8.28 

68.5584 

567.664 

2.87750 

9.09945 

2.02307 

4.35856 

9.39024 

.120773 

8.29 

68.7241 

569.723 

2.87924 

9.10494 

2.02388 

4.36032 

9.39402 

.120627 

8.30 

68.8900 

571.787 

2.88097 

9.11043 

2.02469 

4.36207 

9.39780 

.120482 

8.31 

69.0561 

573.856 

2.88271 

9.11592 

2.02551 

4.36382 

9.40157 

.120337 

8.32 

69.2224 

575.930 

2.88444 

9.12140 

2.02632 

4.36557 

9.40534 

.120192 

8.33 

69.3889 

578.010 

2.88617 

9.12688 

2.02713 

4.36732 

9.40911 

.120048 

8.34 

69.5556 

580.094 

2.88791 

9.13236 

2.02794 

4.36907 

9.41287 

.119904 

8.35 

69.7225 

582.183 

2.88964 

9.13783 

2.02875 

4.37081 

9.41663 

.119761 

8.36 

69.8896 

584.277 

2.89137 

9.14330 

2.02956 

4.37255 

9.42039 

.119617 

8.37 

70.0569 

586.376 

2.89310 

9.14877 

2.03037 

4.37430 

9.42414 

.119474 

8.38 

70.2244 

588.480 

2.89482 

9.15423 

2.03118 

4.37604 

9.42789 

.119332 

8.39 

70.3921 

590.590 

2.89655 

9.15969 

2.03199 

4.37778 

9.43164 

.119190 

8.40 

70.5600 

592.704 

2.89828 

9.16515 

2.03279 

4.37952 

9.43539 

.119048 

8.41 

70.7281 

594.823 

2.90000 

9.17061 

2.03360 

4.38126 

9.43913 

.118906 

8.42 

70.8964 

596.948 

2.90172 

9.17606 

2.03440 

4.38299 

9.44287 

.118765 

8.43 

71.0649 

599.077 

2.90345 

9.18150 

2.03521 

4.38473 

9.44661 

.118624 

8.44 

71.2336 

601.212 

2.90517 

9.18695 

2.03601 

4.38646 

9.45034 

.118483 

8.45 

71.4025 

603.351 

2.90689 

9.19239 

2.03682 

4.38819 

9.45407 

.118343 

8.46 

71.5716 

605.496 

2.90861 

9.19783 

2.03762 

4.38992 

9.45780 

.118203 

8.47 

71.7409 

607.645 

2.91033 

9.20326 

2.03842 

4.39165 

9.46152 

.118064 

8.48 

71.9104 

609.800 

2.91204 

9.20869 

2.03923 

4.39338 

9.46525 

.117925 

8.49 

72.0801 

611.960 

2.91376 

9.21412 

2.04003 

4.39511 

9.46897 

.117786 

8.50 

72.2500 

614.125 

2.91548 

9.21954 

2.04083 

4.39683 

9.47268 

.117647 

POWERS,  ROOTS,  AND  RECIPROCALS.  112m 


„ 

n2 

n:i 

Vi n 

^I^On 

'yin 

<10  n 

<T00n 

1 

n 

8.51 

72.1201 

616.295 

2.91719 

9.22497 

2.04163 

4.39855 

9.47640 

.117509 

8.52 

72.5901 

618.470 

2.91890 

9.23038 

2.04243 

4.40028 

9.48011 

.117371 

8.53 

72.7609 

620.650 

2.92062 

9.23580 

2.04323 

4.40200 

9.48381 

.117233 

8.51 

72.9316 

622.836 

2.92233 

9.24121 

2.04402 

4.40372 

9.48752 

.117096 

8.55 

73.1025 

625.026 

2.92404 

9.24662 

2.04482 

4.40543 

9.49122 

.116959 

8.56 

73.2736 

627.222 

2.92575 

9.25203 

2.04562 

4.40715 

9.49492 

.116822 

8.57 

73.1419 

629.123 

2.92716 

9.25743 

2.04641 

4.40887 

9.49861 

.116686 

8.58 

73.6164 

631.629 

2.92916 

9.26283 

2.04721 

4.41058 

9.50231 

.116550 

8.59 

73.7881 

633.810 

2.90387 

9.26823 

2.04801 

4.41229 

9.50600 

.116414 

8.60 

73.9600 

636.056 

2.93258 

9.27362 

2.04880 

4.41400 

9.50969 

.116279 

8.61 

74.1321 

638.277 

2.93428 

9.27901 

2.04959 

4.41571 

9.51337 

.116144 

8.62 

74.3014 

640.504 

2.93598 

9.28440 

2.05039 

4.41742 

9.51705 

.116009 

8.63 

74.1769 

612.736 

2.93769 

9.28978 

2.05118 

4.41913 

9.52073 

.115875 

8.61 

71.6196 

644.973 

2.93939 

9.29516 

2.05197 

4.42084 

9.52441 

.115741 

8.65 

74.8225 

647.215 

2.94109 

9.30054 

2.05276 

4.42254 

9.52808 

.115607 

8.66 

74.9956 

649.462 

2.94279 

9.30591 

2.05355 

4.42425 

9.53175 

.115473 

8.67 

75.1689 

651.711 

2.94449 

9.31128 

2.05434 

4.42595 

9.53542 

.115340 

8.68 

75.3424 

653.972 

2.94618 

9.31665 

2.05513 

4.42765 

9.53908 

.115207 

8.69 

75.5161 

656.235 

2.94788 

9.32202 

2.05592 

4.42935 

9.54274 

.115075 

8.70 

75.6900 

658.503 

2.94958 

9.32738 

2.05671 

4.43105 

9.54640 

.114943 

8.71 

75.8641 

660.776 

2.95127 

9.33274 

2.05750 

4.43274 

9.55006 

.114811 

8.72 

76.0384 

663.055 

2.95296 

9.33809 

2.05828 

4.43444 

9.55371 

.114679 

8.73 

76.2129 

665.339 

2.95466 

9.34345 

2.05907 

4.43614 

9.55736 

.114548 

8.74 

76.3876 

667.628 

2.95635 

9.34880 

2.05986 

4.43783 

9.56101 

.114417 

8.75 

76.5625 

669.922 

2.95804 

9.35414 

2.06064 

4.43952 

9.56466 

.114286 

8.76 

76.7376 

672.221 

2.95973 

9.35949 

2.06143 

4.44121 

9.56830 

.114155 

8.77 

76.9129 

674.526 

2.96142 

9.36483 

2.06221 

4.44290 

9.57194 

.114025 

8.78 

77.0884 

676.836 

2.96311 

9.37017 

2.06299 

4.44459 

9.57557 

.113895 

8.79 

77.2641 

679.151 

2.96479 

9.37550 

2.06378 

4.44627 

9.57921 

.113766 

8.80 

77.4400 

681.472 

2.96648 

9.38083 

2.06456 

4.44796 

9.58284 

.113636 

8.81 

77.6161 

683.798 

2.96816 

9.38616 

2.06534 

4.44964 

9.58647 

.113507 

8.82 

77.7924 

686.129 

2.96985 

9.39149 

2.06612 

4.45133 

9.59009 

.113379 

8.83 

77.9689 

688.465 

2.97153 

9.39681 

2.06690 

4.45301 

9.59372 

.113250 

8.84 

78.1456 

690.807 

2.97321 

9.40213 

2.06768 

4.45469 

9.59734 

.113122 

8.85 

78.3225 

•693.154 

2.97489 

9.40744 

2.06846 

4.45637 

9.60095 

.112994 

8.86 

78.4996 

695.506 

2.97658 

9.41276 

2.06924 

4.45805 

9.60457 

.112867 

8.87 

78.6769 

697.864 

2.97825 

9.41807 

2.07002 

4.45972 

9.60818 

.112740 

8.88 

78.8544 

700.227 

2.97993 

9.42338 

2.07080 

4.46140 

9.61179 

.112613 

8.89 

79.0321 

702.595 

2.98161 

9.42868 

2.07157 

4.46307 

9.61540 

.112486 

8.90 

79.2100 

704.969 

2.98329 

9.43398 

2.07235 

4.46474 

9.61900 

.112360 

8.91 

79.3881 

707.348 

2.98496 

9.43928 

2.07313 

4.46642 

9.62260 

.112233 

8.92 

79.5664 

709.732 

2.98664 

9.44458 

2.07390 

4.46809 

9 62620 

.112108 

8.93 

79;  7449 

712.122 

2.98831 

9.44987 

2.07468 

4.46976 

9.62980 

.111982 

8.94 

79.9236 

714.517 

2.98998 

9.45516 

2.07545 

4.47142 

9.63339 

.111857 

8.95 

80.1025 

716.917 

2.99166 

9.46044 

2.07622 

4.47309 

9.63698 

.111732 

8.96 

80.2816 

719.323 

2.99333 

9.46573 

2.07700 

4.47476 

9.64057 

.111607 

8.97 

80.4609 

721.734 

2.99500 

9.47101 

2.07777 

4.47642 

9.64415 

.111483 

8.98 

80.6404 

724.151 

2.99666 

9.47629 

2.07854 

4.47808 

9.64774 

.111359 

8.99 

80.8201 

726.573 

2.99833 

9.48156 

2.07931 

4.47974 

9.65132 

.111235 

9.00 

81.0000 

729.000 

3.00000 

9.48683 

2.08008 

4.48140 

9.65489 

.111111 

112 ft  POWERS,  ROOTS,  AND  RECIPROCALS. 


n 

n* 

n 3 

VIo Jl 

tin 

AlO  n 

"VlOOn 

1 

n 

9.01 

81.1801 

731.433 

3.00167 

9.49210 

2.08085 

4.48306 

9.65847 

.110988 

9.02 

81.3604 

733.871 

3.00333 

9.49737 

2.08162 

4.48472 

9.66204 

.110865 

9.03 

81.5409 

736.314 

3.00500 

9.50263 

2.08239 

4.48638 

9.66561 

.110742 

9.04 

81.7216 

738.763 

3.00666 

9.50789 

2.08316 

4.48803 

9.66918 

.110620 

9.05 

81.9025 

741.218 

3.00832 

9.51315 

2.08393 

4.48968 

9.67274 

.110497 

9.06 

82.0836 

743.677 

3.00998 

9.51840 

2.08470 

4.49134 

9.67630 

.110375 

9.07 

82.2649 

746.143 

3.01164 

9.52365 

2.08546 

4.49299 

9.67986 

.110254 

9.08 

82.4464 

748.613 

3.01330 

9.52890 

2.08623 

4.49464 

9.68342 

.110132 

9.09 

82.6281 

751.089 

3.01496 

9.53415 

2.08699 

4.49629 

9.68697 

.110011 

9.10 

82.8100 

753.571 

3.01662 

9.53939 

2.08776 

4.49794 

9.69052 

.109890 

9.11 

82.9921 

756.058 

3.01828 

9.54463 

2.08852 

4.49959 

9.69407 

.109770 

9.12 

83.1744 

758.551 

3.01993 

9.54987 

2.08929 

4.50123 

9.69762 

.109649 

9.13 

83.3569 

761.048 

3.02159 

9.55510 

2.09005 

4.50288 

9.70116 

.109529 

9.14 

83.5396 

763.552 

3.02324 

9.56033 

2.09081 

4.50452 

9.70470 

.109409 

9.15 

83.7225 

766.061 

3.02490 

9.56556 

2.09158 

4.50616 

9.70824 

.109290 

9.16 

83.9056 

768.575 

3.02655 

9.57079 

2.09234 

4.50780 

9.71177 

.109170 

9.17 

84.0889 

771.095 

3.02820 

9.57601 

2.09310 

4.50945 

9.71531 

.109051 

9.18 

84.2724 

773.621 

3.02985 

9.58123 

2.09386 

4.51108 

9.71884 

.108933 

9.19 

84.4561 

776.152 

3.03150 

9.58645 

2.09462 

4.51272 

9.72236 

.108814 

9.20 

84.6400 

778.688 

3.03315 

9.59166 

2.09538 

4.51436 

9.72589 

.108696 

9.21 

84.8241 

781.230 

3.03480 

9.59687 

2.09614 

4.51599 

9.72941 

.108578 

9.22 

85.0084 

783.777 

3.03645 

9.60208 

2.09690 

4.51763 

9.73293 

.108460 

9.23 

85.1929 

786.330 

3.03809 

9.60729 

2.09765 

4.51926 

9.73645 

.108342 

9.24 

85.3776 

788.889 

3.03974 

9.61249 

2.09841 

4.52089 

9.73996 

.108225 

9.25 

85.5625 

791.453 

3.04138 

9.61769 

2.09917 

4.52252 

9.74348 

.108108 

9.26 

85.7476 

794.023 

3.04302 

9.62289 

2.09992 

4.52415 

9.74699 

.107991 

9.27 

85.9329 

796.598 

3.04467 

9.62808 

2.10068 

4.52578 

9.75049 

.107875 

9.28 

86.1184 

799.179 

3.04631 

9.63328 

2.10144 

4.52740 

9.75400 

.107759 

9.29 

86.3041 

801.765 

3.04795 

9.63846 

2.10219 

4.52903 

9.75750 

.107643 

9.30 

86.4900 

804.357 

3.04959 

9.64365 

2.10294 

4.53065 

9.76100 

.107527 

9.31 

86.6761 

806.954 

3.05123 

9.64883 

2.10370 

4.53228 

9.76450 

.107411 

9.32 

86.8624 

809.558 

3.05287 

9.65401 

2.10445 

4.53390 

9.76799 

.107296 

9.33 

87.0489 

812.166 

3.05450 

9.65919 

2.10520 

4.53552 

9.77148 

.107181 

9.34 

87.2356 

814.781 

3.05614 

9.66437 

2.10595 

4.53714 

9.77497 

.107066 

9.35 

87.4225 

817.400 

3.05778 

9.66954 

2.10671 

4.53876 

9.77846 

.106952 

9.36 

87.6096 

820.026 

3.05941 

9.67471 

2.10746 

4.54038 

9.78195 

.106838 

9.37 

87.7969 

822.657 

3.06105 

9.67988 

2.10821 

4.54199 

9.78543 

.106724 

9.38 

87.9844 

825.294 

3.06268 

9.68504 

2.10896 

4.54361 

9.78891 

.106610 

9.39 

88.1721 

827.936 

3.06431 

9.69020 

2.10971 

4.54522 

9.79239 

.106496 

9.40 

88.3600 

830.584 

3.06594 

9.69536 

2.11045 

4.54684 

9.79586 

.106383 

9.41 

88.5481 

833.238 

3.06757 

9.70052 

2.11120 

4.54845 

9.79933 

.106270 

9.42 

88.7364 

835.897 

3.06920 

9.70567 

2.11195 

4.55006 

9.80280 

.106157 

9.43 

88.9249 

838.562 

3.07083 

9.71082 

2.11270 

4.55167 

9.80627 

.106045 

9.44 

89.1136 

841.232 

3.07246 

9.71597 

2.11344 

4.55328 

9.80974 

.105932 

9.45 

89.3025 

843.909 

3.07409 

9^72111 

2.11419 

4.55488 

9.81320 

.105820 

9.46 

89.4916 

846.591 

3.07571 

9.72625 

2.11494 

4.55649 

9.81666 

.105708 

9.47 

89.6809 

849.278 

3.07734 

9.73139 

2.11568 

4.55809 

9.82012 

.105597 

9.48 

89.8704 

851.971 

3.07896 

9.73653 

2.11642 

4.55970 

9.82357 

.105485 

9.49 

90.0601 

854.670 

3.08058 

9.74166 

2.11717 

4.56130 

9.82703 

.105374 

9.50 

90.2500 

857.375 

3.08221 

9.74679 

2.11791 

4.56290 

9.83048 

.105263 

POWERS,  ROOTS,  AND  RECIPROCALS.  112 0 


n 

n2 

n3 

Vn 

A 

© 

3 1 

$100n 

1 

n 

9.51 

90.4401 

860.085 

3.08383 

9.75192 

2.11865 

4.56450 

9.83392 

.105153 

9.52 

90.6304 

862.801 

3.08545 

9.75705 

2.11940 

4.56610 

9.83737 

.105042 

9.53 

90.8209 

865.523 

3.08707 

9.76217 

2.12014 

4.56770 

9.84081 

.104932 

9.54 

91.0116 

868.251 

3.08869 

9.76729 

2.12088 

4.56930 

9.84425 

.104822 

9.55 

91.2025 

870.984 

3.09031 

9.77241 

2.12162 

4.57089 

9.84769 

.104712 

9.56 

91.3936 

873.723 

3.09192 

9.77753 

2.12236 

4.57249 

9.85113 

.104603 

9.57 

91.5849 

876.467 

3.09354 

9.78264 

2.12310 

4.57408 

9.85456 

.104493 

9.58 

91.7764 

879.218 

3.09516 

9.78775 

2.12384 

4.57568 

9.85799 

.104384 

9.59 

91.9681 

881.974 

3.09677 

9.79285 

2.12458 

4.57727 

9.86142 

.104275 

9.60 

92.1600 

884.736 

3.09839 

9.79796 

2.12532 

4.57886 

9.86485 

.104167 

9.61 

92.3521 

887.504 

3.10000 

9.80306 

2.12605 

4.58045 

9.86827 

.104058 

9.62 

92.5444 

890.277 

3.10161 

9.80816 

2.12679 

4.58203 

9.87169 

.103950 

9.63 

92.7369 

893.056 

3.10322 

9.81326 

2.12753 

4.58362 

9.87511 

.103842 

9.64 

92.9296 

895.841 

3.10483 

9.81835 

2.12826 

4.58521 

9.87853 

.103734 

9.65 

93.1225 

898.632 

3.10644 

9.82344 

2.12900 

4.58679 

9.88195 

.103627 

9.66 

93.3156 

901.429 

3.10805 

9.82853 

2.12974 

4.58838 

9.88536 

.103520 

9.67 

93.5089 

904.231 

3.10966 

9.83362 

2.13047 

4.58996 

9.88877 

.103413 

9.68 

93.7024 

907.039 

3.11127 

9.83870 

2.13120 

4.59154 

9.89217 

.103306 

9.69 

93.8961 

909.853 

3.11288 

9.84378 

2.13194 

4.59312 

9.89558 

.103199 

9.70 

94.0900 

912.673 

3.11448 

9.84886 

2.13267 

4.59470 

9.89898 

.103093 

9.71 

94.2841 

915.499 

3.11609 

9.85393 

2.13340 

4.59628 

9.90238 

.102987 

9.72 

94.4784 

918.330 

3.11769 

9.85901 

2.13414 

4.59786 

9.90578 

.102881 

9.73 

94.6729 

921.167 

3.11929 

9.86408 

2J3487 

4.59943 

9.90918 

.102775 

9.74 

94.8676 

924.010 

3.12090 

9.86914 

2.13560 

4.60101 

9.91257 

.102669 

9.75 

95.0625 

926.859 

3.12250 

9.87421 

2.13633 

4.60258 

9.91596 

.102564 

9.76 

95.2576 

929.714 

3.12410 

9.87927 

2.13706 

4.60416 

9.91935 

.102459 

9.77 

95.4529 

932.575 

3.12570 

9.88433 

2.13779 

4.60573 

9.92274 

.102354 

9.78 

95.6484 

935.441 

3.12730 

9.88939 

2.13852 

4.60730 

9.92612 

.102250 

9.79 

95.8441 

938.314 

3.12890 

9.89444 

2.13925 

4.60887 

9.92950 

.102145 

9.80 

96.0400 

941.192 

3.13050 

9.89949 

2.13997 

4.61044 

9.93288 

.102041 

9.81 

96.2361 

944.076 

3.13209 

9.90454 

2.14070 

4.61200 

9.93626 

.101937 

9.82 

96.4324 

946.966 

3.13369 

9.90959 

2.14143 

4.61357 

9.93964 

.101833 

9.83 

96.6289 

949.862 

3.13528 

9.91464 

2.14216 

4.61513 

9.94301 

.101729 

9.84 

96.8256 

952.764 

3.13688 

9.91968 

2.14288 

4.61670 

9.94638 

.101626 

9.85  i 

.97.0225 

955.672 

3.13847 

9.92472 

2.14361 

4.61826 

9.94975 

.101523 

9.86 

97.2196 

958.585 

3.14006 

9.92975 

2.14433 

4.61983 

9.95311 

.101420 

9.87 

97.4169 

961.505 

3.14166 

9.93479 

2.14506 

4.62139 

9.95648 

.101317 

9.88 

97.6144 

964.430 

3.14325 

9.93982 

2.14578 

4.62295 

9.95984 

.101215 

9.89 

97.8121 

967.362 

3.14484 

9.94485 

2.14651 

4.62451 

9.96320 

.101112 

9.90 

98.0100 

970.299 

3.14643 

9.94987 

2.14723 

4.62607 

9.96655 

.101010 

9.91 

98.2081 

973.242 

3.14802 

9.95490 

2.14795 

4.62762 

9.96991 

.100908 

9.92 

98.4064 

976.191 

3.14960 

9.95992 

2.14867 

4.62918 

9.97326 

.100807 

9.93 

98.6049 

979.147 

3.15119 

9.96494 

2.14940 

4.63073 

9.97661 

.100705 

9.94 

98.8036 

982.108 

3.15278 

9.96995 

2.15012 

4.63229 

9.97996 

.100604 

9.95 

99.0025 

985.075 

3.15436 

9.97497 

2.15084 

4.63384 

9.98331 

.100503 

9.96 

99.2016 

988.048 

3.15595 

9.97998 

2.15156 

4.63539 

9.98665 

.100402 

9.97 

99.4009 

991.027 

3.15753 

9.98499 

2.15228 

4.63694 

9.98999 

.100301 

9.98 

99.6004 

994.012 

3.15911 

9.98999 

2.15300 

4.63849 

9.99333 

.100200 

9.99 

99.8001 

997.003 

3.16070 

9.99500 

2.15372 

4.64004 

9.99667 

.100100 

10.00 

100.000 

1000.00 

3.16228 

10.0000 

2.15443 

4.64159 

10.0000 

.100000 

112 p 


DECIMAL  EQUIVALENTS. 


DECIMAL  EQUIVALENTS  OF  64ths. 

The  decimal  fractions  printed  in  large  type  give  the  exact 
value  of  the  corresponding  fraction  to  the  fourth  decimal 
place.  A given  decimal  fraction  is  rarely  exactly  equal  to 
any  of  these  values,  and  the  numbers  in  small  type  show 
which  common  fraction  is  nearest  to  the  given  decimal. 
Thus,  lay  off  the  fraction  .1330  in  64ths.  The  nearest  decimal 
fractions  are  .1250  and  .1406.  The  value  of  any  fraction  in 
small  type  is  the  mean  of  the  two  adjacent  fractions.  In 
this  instance  the  mean  fraction  is  .1328,  and  as  .1330  is  greater 
than  this,  .1406  or  st  will  be  chosen.  In  the  same  manner 
the  nearest  64ths  corresponding  to  the  decimal  fractions  .3670 
and  .8979  are  found  to  be  §f  and  §£,  respectively. 


Frac- 

tion 

Decimal 

Frac- 

tion 

Decimal 

Frac- 

tion 

Decimal 

Frac- 

tion 

Decimal 

.0078 

.2578 

.5078 

.7578 

A 

.0156 

.2656 

M 

.5156 

If 

.7656 

.0235 

.2735 

.5235 

.7735 

A 

.0313 

A 

.2813 

17 

ST 

.5313 

if 

.7813 

.0391 

.2891 

.5391 

.7891 

A 

.0469 

if 

.2969 

If 

.5469 

If 

.7969 

.0547 

.3047 

.5547 

.8047 

A 

.0625 

TS 

.3125 

TS 

.5625 

H 

.8125 

.0703 

.3203 

.5703 

.8203 

& 

.0781 

a 

.3281 

H 

.5781 

« 

.8281 

.0860 

.3360 

.5860 

.8360 

A 

.0938 

a 

.3438 

if 

.5938 

if 

.8438 

.1016 

.3516 

.6016 

.8516 

A 

.1094 

ii 

.3594 

If 

.6094 

If 

.8594 

.1172 

.3672 

6172 

.6672 

£ 

.1250 

I 

.3750 

5 

s 

.6250 

7 

s 

.8750 

.1328 

.3828 

.6328 

.8828 

A 

.1406 

if 

.3906 

If 

.6406 

67 

ST 

.8906 

.1485 

.3985 

.6485 

.8985 

A 

.1563 

if 

.4063 

21 

ST 

.6563 

if 

.9063 

.1641 

4141 

.6641 

.9141 

if 

.1719 

if 

.4219 

If 

.6719 

if 

.9219 

.1797 

.4297 

.6797 

.9297  | 

A 

.1875 

TS 

.4375 

« 

.6875 

ft 

.9375 

.1953 

.4453 

.6953 

.9453 

if 

.2031 

if 

.4531 

If 

.7031 

If 

.9531 

.2110 

.4610 

.7110 

.9610 

A 

.2188 

if 

.4688 

if 

.7188 

If 

.9688 

.2266 

.4766 

.7266 

.9766 

if 

.2344 

If 

.4844 

If 

.7344 

if 

.9844 

.2422 

.4922 

.7422 

.9922 

i 

.2500 

4 

.5000 

f 

.7500 

l 

1.0000 

.2578 

.5078 

.7578 

1.0078 

MENSURATION. 


113 


MENSURATION. 


In  the  following  formulas,  the  letters  have  the  meanings 
here  given,  unless  otherwise  stated. 

D = larger  diameter; 
d — smaller  diameter; 

R = radius  corresponding  to  D; 
r = radius  corresponding  to  d; 
p = perimeter  or  circumference; 

C = area  of  convex  surface  = area  of  flat  surface  which  can 
be  rolled  into  the  shape  shown; 

S = area  of  entire  surface  = C + area  of  the  end  or  ends; 

A = area  of  plane  figure; 

n ==  3.1416,  nearly  = ratio  of  any  circumference  to  its  diam- 
eter; 

V = volume  of  solid. 

The  other  letters  used  will  be  found  on  the  cuts. 


P 

P 

P 

P 

d 

d 

r 

r 

A 

A 

A 


Cl  RCLE. 


ir  d = 3.1416  d. 

2 7t  r = 6.2832  r. 

2 i /VA  = 3.5449  l/A. 
2 A 4 A 


r d 

P P . 

7T  3.1416 


= .3183  p. 


iV  ~ = l.: 


1284  V A. 


P_ 

2 7T 


P 

6.2832 


= .1592 p. 


V - = .5642  V A. 

7 r 


77^2 

4 


.7854  d2. 


7rr2=  3.1416  r2. 
pr  _ pd 

~2  ~ T* 


114 


MENSURATION. 


For 
hypotenuse, 


TRIANGLES. 

D = B+  C.  E + B + C — 180°. 

B = D—C.  E'+  B + C = 180°. 

E'=  E.  B'=  B. 

The  above  letters  refer  to  angles, 
right-angled  triangle,  c being  the 


c = i/a2  + 62. 
a = j/  c2  — 62. 
6 = j/  c2 


a2. 

c = length  of  side  opposite  an  acute  angle 
of  an  oblique-angled  triangle. 

c = |/a2-f  62—  266. 

6 = y/ a2—  e2. 

c = length  of  side  opposite  an  obtuse  angle 
of  an  oblique-angled  triangle. 

c = j/  a2  + 62  + 2 b e. 
h = \/  a2  — e2. 

For  a triangle  inscribed  in  a semicircle;  i.  e.,  any  right- 
angled  triangle, 

c:b::a:h. 

^ _ qfr  _ ce 
~~  c ~~  a * 
a : 6 + e = e:a  = 6 : c. 

For  any  triangle, 

4 = _ = i6A. 


A = 


6 / 
2^“ 


/ a2  + 62  — c2  \ 2 

V 26  j ' 


RECTANGLE  AND  PARALLELOGRAM. 

A = a 6. 

A = 6 |/  c2  — 62. 


MENSURATION. 


115 


TRAPEZOID. 

A = \ ;h{0/-{-b). 


TRAPEZIUM. 

Divide  into  two  triangles  and  a trapezoid. 

A = %bh'+%a(h'+h)  + ±chi 
or,  A = i[6/i'+cA  + a(^'+A)]. 

Or,  divide  into  two  triangles  by  drawing 
a diagonal.  Consider  the  diagonal  as  the 
base  of  both  triangles,  call  its  length  l ; 
call  the  altitudes  of  the  triangles  \ and  then 
A — % l {hi  -j-  ho). 


I— — H) 


ELLI  PSE. 

p*  = T l^  + d2 

\ 2 8.8  * 

A = -A  D d = .7854  D d. 

4 


SECTOR. 

A = ilr. 

7 r 7*2 

A = — = .008727  r2  E. 
ooO 

Z = length  of  arc. 


SEGM  ENT. 

A = i [fr  — c (r  — A)]. 


* The  perimeter  of  an  ellipse  cannot  be  exactly  determined 
without  a very  elaborate  calculation,  and  this  formula  is 
merely  an  approximation  giving  fairly  close  results. 


116 


MENSURATION. 


RING. 

A = j(D*-d*). 


CHORD. 

c = length  of  chord. 

c2  + 4 h1  _ e2 
r = 8/T~  = 2A 

c = 2]/2/ir-/(2. 

J C,  approximately. 


HELIX. 


To  construct  a ’helix. 

I = length  of  helix; 
n — number  of  turns; 
t = pitch. 


t ----- 

l = 
n = 


n ;/  7T2  d2  + t 2. 
I 


Y TV*  a2  + t* 


CYLINDER. 

C = 7rd/i. 

£ = 2tt  r/i  + 2?r  r2 
= Trd/i  + ^d2. 


^ = .0796  p2  A. 

4 7T 


FRUSTUM  OF  CYLINDER. 

h = h sum  of  greatest  and  least  heights. 

c = ph  = tt  d A 

5 = ir  dh  +^d2  area  of  elliptical  top. 

V = Ah  = ydVi.  ' 

4 


MENSURATION. 


117 


CONE. 

C n dl  = it  rl. 

it  v l ~f*  7r  r2  = 7T  r |/  r2  + h?  -f-  n r2. 

r - !L^!  v - - -7854  h _ Pi? 
K_4X3“  3 “ 12n 


FRUSTUM  OF  CONE. 

C=  U(P+p)  = d), 

s = ^[l(D  + d)+UD2+d*)]. 

V=  J(2P  + Dd  + d*)X}h 
= .2618  h (D2  + Dd  -\-  d2). 


SPHERE. 

S=nd2  = 4:Trr2  = 12.5664  r2. 

V = £ tt  d3  ==  1 7T  r3  = .5236  d3  = 4.1888  r3. 


circular  ring. 

JX  = mean  diameter; 

B = mean  radius. 

S = 4 7T2  E r = 9.8696  D d. 
V = 2 7T2  r2  = 2.4674  D d2. 


WEDGE. 

F = h(  a + 5 + c). 


PRISMOID. 

A prismoid  is  a solid  having  two  parallel  plane  ends,  the 
edges  of  which  are  connected  by  plane  triangular  or  quadri- 
lateral surfaces. 


118 


MENSURATION. 


A = area  one  end; 
a = area  of  other  end; 
m = area  of  section  midway  between  ends; 
l = perpendicular  distance  between  ends. 
V = l l(A  -f  a + 4m). 

The  area  m is  not  in  general  a mean  between  the  areas  of 
the  two  ends,  but  its  sides  are  means  between  the  correspond- 
ing lengths  of  the  ends. 

Approximately,  V = l- 


REGULAR  PYRAMID. 

P = perimeter  of  base; 

A = area  of  base. 

C = kPl- 
S = % PI  + A. 

V = AA 

3 ' 

To  obtain  area  of  base,  divide  it  into  triangles,  and  find 
their  sum. 

The  formula  for  V applies  to  any  pyramid  whose  base  is 
A and  altitude  h. 


FRUSTUM  OF  REGULAR  PYRAMID. 

a = area  of  upper  base; 

A = area  of  lower  base; 
p = perimeter  of  upper  base; 

P = perimeter  of  lower  base. 

C=  U(P  + p). 

S = $ l ( P + p ) -\-A  + cl 
V = ±h(A  + a+\/Aa). 

The  formula  for  V applies  to  the  frustum  of  any  pyramid. 


I = 
l = 


LENGTH  OF  SPIRAL. 

( D + d\  n — number  of  coil; 

\ 2 / * l ==  length  of  spiral; 

~ ( P2  — ) . t = pitch. 


MENSURATION. 


119 


PRISM  OR  PARALLELOPIPED 

C = Ph. 

S = Ph  + 2A. 

V = Ah.  A 

For  prisms  with  regular  polygon  as 
bases,  P = length  of  one  side  X number  of  sides. 

To  obtain  area  of  base,  if  it  is  a polygon,  divide  it  into  tri- 
angles, and  find  sum  of  partial  areas. 


FRUSTUM  OF  PRISM. 

If  a section  perpendicular  to  the  edges  is  a 
triangle,  square,  parallelogram,  or  regular  polygon, 
sum  of  lengths  of  edges 


V = 
section. 


number  of  edges 


X area  of  right 


REGULAR  POLYGONS. 

Divide  the  polygon  into  equal  triangles  and  find  the  sum  of 
the  partial  areas.  Otherwise,  square  the  length  of  one  side 
and  multiply  by  proper  number  from  the  following  table: 
Name.  No.  Sides.  Multiplier . 


Triangle 

3 

.433 

Square 

4 

1.000 

Pentagon 

5 

1.720 

Hexagon 

6 

2.598 

Heptagon 

7 

3.634 

Octagon 

8 

4.828 

Nonagon 

9 

6.182 

Decagon 

10 

7.694 

IRREGULAR  AREAS. 

Divide  the  area  into  trapezoids,  triangles,  parts 
of  circles,  etc.,  and  find  the  sum  of  the  partial  areas. 

If  the  figure  is  very  irregular,  the  approximate 
area  may  be  found  as  follows : Divide  the  figure 
into  trapezoids  by  equidistant  parallel  lines  b,  c,  d, 
etc.  The  lengths  of  these  lines  being  measured, 
then,  calling  a the  first  and  n the  last  length,  and 
y the  width  of  strips, 

Area  ==  y — + b + c + etc .+ . 


120 


MECHANICS. 


MECHANICS. 


FALLING  BODIES. 

Let  g = 32.16  = constant  acceleration  due  to  the  attrac- 
tion of  the  earth; 

t = number  of  seconds  that  the  body  falls; 

v = velocity  in  feet  per  second  at  the  end  of  the 
time  t; 

h — distance  that  the  body  falls  during  the  time  t. 

Then,  v = g t = ^ = y/2gh  = 8.02  j/  h. 

* = T = 015547  ^ 

t = - = — = -J—  = .24938  \/~K 
g v \ g 

PROJECTILES. 

The  formulas  under  this  and  the  preceding  heading  are 
rigidly  true  only  for  bodies  moving  in  a vacuum  or  in  space 
(as  the  stars  and  planets);  they  are  approximately  true  for 
bodies  moving  in  air,  provided  they  are  dense  and  the 
velocity  is  not  very  great.  Fairly  good  results  may  be  ob- 
tained by  applying  the  formulas  for  projectiles  in  calculating 
the  range  of  a jet  of  water  issuing  from  a small  orifice  in  the 
side  of  a vessel. 

Let  g = 32.16  = acceleration  due  to  gravity; 

v = initial  velocity  in  feet  per  second; 

r = range; 

y = vertical  height  of  starting  point  above  ground; 

A = elevation  in  degrees  = angle  that  the  direction 
of  the  projectile  at  the  start  makes  with  the 
horizontal. 

Then  the  range,  or  distance  from  the  starting  point  to  the 
point  where  the  projectile  crosses  a horizontal  line  through 
the  starting  point,  is 

qfi 

r = — sm  2 A. 

9 


CENTER  OF  GRAVITY. 


121 


If  the  body  is  projected  in  a horizontal  direction,  the  range 
is  the  distance  from  the  starting  point  to  the  point  where  the 
projectile  strikes  the  ground,  and 

r = = -24938  v\ /y. 

The  range  of  a projectile  fired  in  a horizontal  direction, 
30  ft.  above  the  ground,  with  a velocity  of  300  ft.  per  second, 
equals  r = .24938  X 300  X V 30  = 409.77  ft. 


CENTRIFUGAL  FORCE. 

F = centrifugal  force  in  pounds; 

W=  w eight  of  revolving  body  in  pounds; 

r = distance  from  the  axis  of  motion  to  the  center  of 
gravity  of  the  body  in  feet; 

N = number  of  revolutions  per  minute; 

v = velocity  in  feet  per  second. 

Wv2 

F = — — = .00034  Wr  N2. 

gr 

In  calculating  the  centrifugal  force  of  flywheels,  it  is 
customary  to  neglect  the  arms  and  take  r equal  to  the  mean 
radius  of  the  rim;  in  such  cases  W is  taken  as  one-half  the 
weight  of  the  rim.  The  result  thus  obtained,  divided  by  n,  is 
approximately  the  force  tending  to  burst  the  flywheel  rim. 

Example.— What  is  the  force  tending  to  burst  a flywheel 
rim  weighing  7 tons,  making  150  rev.  per  min.,  and  having 
a mean  radius  of  5 ft.? 

Solution.— 

_ .00034  X (I  X 7 X 2,000  ) 5 X 1502  _ _ 997 
F 3 .1416  85,227  lb* 


CENTER  OF  GRAVITY. 

The  center  of  gravity  of  a body,  or  of  a system  of  bodies,  is 
that  point  from  which,  if  the  body  or  system  were  suspended, 
it  would  be  in  equilibrium. 

If  a line  or  a surface  has  two  axes,  or  a solid  has  three  axes 
of  symmetry,  the  center  of  gravity  lies  at  their  point  of  inter- 
section, and  corresponds  with  the  geometrical  center  of  the 
figure. 


122 


MECHANICS. 


An  axis  of  symmetry  is  any  line  so  drawn  that,  if  part  of 
the  figure  on  one  side  of  the  line  is  folded  on  this  line,  it  will 
coincide  exactly  with  the  other  part,  point  for  point  and  line 
for  line.  Thus,  in  Fig.  1,  if  the  part  a b is  folded  on  the  line 
A B,  the  upper  half  will  coincide  exactly  with  the  lower 
half;  also,  if  be  is  folded  on  the  line  CD,  the  right-hand  half 
will  coincide  exactly  with  the  left-hand  half.  Hence,  the 
point  0 where  A B and  C D intersect  is  the  center  of  gravity 
of  the  rectangle  abed.  If  the  figure  has  one  axis  of 
symmetry,  the  center  of  gravity  may  he  found  as  follows:  Let 


m n be  an  axis  of  symmetry  of  the  area  in  Fig.  2.  The  center 
of  gravity  will  lie  somewhere  on  this  line.  Draw  any  line  A B 
perpendicular  to  mn.  Divide  the  area  into  squares,  rect- 
angles, triangles,  parallelograms,  circles,  etc.,  whose  centers 
of  gravity  are  easily  found,  and  measure  the  perpendicular 
distances  of  these  centers  of  gravity  from,  the  line  A B.  Add 
the  sum  of  the  products  obtained  by  multiplying  each  area 
by  the  distance  of  its  center  of  gravity  from  the  line  A B,  and 
divide  by  the  area  of  the  entire  figure;  the  result  is  the  dis- 
tance x of  the  center  of  gravity  from  A B measured  on  m n , 
^r  the  point  F. 

If  the  figure  has  no  axis  of  symmetry,  as  in  Fig.  3,  draw  any 
line,  as  A B,  and  find  the  distance  x of  the  center  of  gravity 
from  A B,  and  through  x draw  fg  parallel  to  A B.  Choose  any 
other  line,  CD,  and  find  the  distance  y of  the  center  of  gravity 
from  CD  by  the  same  method,  and  through  y draw  mn 
parallel  to  C D.  The  point  of  intersection  o of  / g and  mn  is 
the  center  of  gravity. 

Thus,  suppose  that  the  area  of  the  triangle,  Fig.  3,  is 
A sq.  in.,  and  the  distance  of  its  center  of  gravity  from  A B is 


CENTER  OF  GRAVITY. 


123 


a in.,  and  from  C D , a\  in.;  that  the  area  of  the  small  rectangle 
is  B sq.  in.,  and  the  distance  of  its  center  of  gravity  from  A B 
is  b in.,  and  from  C D is  bi  in.;  that  the  area  of  the  large  rect- 
angle is  C sq.  in.,  and  the  distance  of  its  center  of  gravity 
from  A B is  c in.,  and  from  CD  is  C\  in.;  then, 

_ (iXo)  + (BXb)  + (CXc) 

X ~ A + B + C 

* UXcq)  + ( BXh ) + (tfXCi) 

and  y = I+TTC • 

To  find  the  center  of  gravity  mechanically,  suspend  the 
object  from  a point  near  its  edge  and  mark  on  it  the  direction 
of  a plumb-line  from  that  point;  then  suspend  it  from  another 
point  and  'again  mark  the  direction  of  a plumb-line.  The 
intersection  of  these  two  lines  will  be  directly  over  the  center 
of  gravity. 

The  center  of  gravity  of  a body  having  parallel  sides  may 
be  found  by  drawing  the  outline  of  one  of  the  sides  upon 
heavy  paper,  and  cutting  out  the  exact  shape  of  the  figure. 
Then  suspend  the  paper  from  the  two  points  and  find  the 
center  of  gravity,  as  in  the  last  case. 

The  center  of  gravity  of  a triangle  lies  on  a line  drawn 
from  a vertex  to  the  middle  point  of  the  opposite  side,  and 
at  a distance  from  that  side  equal  to  one-third  of  the  length 
of  the  line.  Or,  draw  a line  from  another  vertex  to  the 
middle  point  of  the  side  opposite,  and  the  intersection  of  the 
two  lines  will  be  the  center  of  gravity. 

For  a parallelogram,  the  center  of  gravity  is  at  the  inter- 
section of  the  two  diagonals. 

For  an  irregular  four-sided  figure,  draw  a diagonal,  divi- 
ding it  into  two  triangles.  Draw  a line  joining  these  centers 
of  gravity.  Draw  the  other  diagonal,  dividing  the  figure 
into  two  other  triangles,  and  join  the  centers  of  gravity  by  a 
straight  line.  The  intersection  of  these  lines  is  the  center 
of  gravity  of  the  figure. 

For  a figure  having  more  than  four  sides,  find  the  center 
of  gravity  by  the  general  method  explained  in  connection 
with  Fig.  3. 

For  an  arc  of  a circle,  the  center  of  gravity  lies  on  the 
radius  drawn  to  the  middle  point  of  the  arc  (an  axis  ol 


124 


MECHANICS. 


symmetry)  and  at  a distance  from  the  center  equal  to  the 
length  of  the  chord  multiplied  by  the  radius  and  divided 
by  the  length  of  the  arc. 

2 

For  a semicircle,  the  distance  from  the  center  = — 

7T 

= .6366  r,  when  r = the  radius. 

For  the  area  included  in  a half  circle,  the  distance  of  the 

center  of  gravity  from  the  center  =#  ~ = .4244  r. 

6 7 r 

For  circular  sector,  the  distance  of  the  center  of  gravity 
from  the  center  equals  two-thirds  of  the  length  of  the  chord 
multiplied  by  the  radius  and  divided  by  the  length  of  the  arc. 

For  a circular  segment,  let  A be  its  area  and  C the  length 
of  its  chord;  then  the  distance  of  the  center  of  gravity  from 
C3 

the  center  of  the  circle  is  equal  to 

For  a solid  having  three  axes  of  symmetry,  all  perpen- 
dicular to  each  other,  like  a sphere,  cube,  right  parallelo- 
piped,  etc.,  their  point  of  intersection  is  the  center  of  gravity. 

For  a cone  or  pyramid,  draw  a line  from  the  apex  to  the 
center  of  gravity  of  the  base;  the  required  center  of  gravity 
is  one-fourth  the  length  of  this  line  from  the  base,  measured 
on  the  line. 

For  two  bodies,  the  larger  weighing  W lb.,  and  the  smaller 
P lb.,  the  center  of  gravity  will  lie  on  the  line  joining  the 
centers  of  gravity  of  the  two  bodies  and  at  a distance  from  the 
Pa 

larger  body  equal  to  p + ^ , where  a is  the  distance  between 

the  centers  of  gravity  of  the  two  bodies. 

For  any  number  of  bodies,  first  find  the  center  of  gravity  of 
two  of  them  as  above,  and  consider  them  as  one  weight  whose 
center  of  gravity  is  at  the  point  just  found.  Find  the  center 
of  gravity  of  this  combined  weight  and  a third  body.  So 
continue  for  the  rest  of  the  bodies,  and  the  last  center  of 
gravity  will  be  the  center  of  gravity  of  the  whole  system  of 
bodies. 


MOMENT  OF  INERTIA. 


125 


MOMENT  OF  INERTIA. 

The  moment  of  inertia  of  a body  or  section  is  a mathematical 
expression  that  is  much  used  in  computations  relating  to 
rotating  bodies  and  to  the  strength  of  materials. 

It  may  be  defined  as  follows: 

The  moment  of  inertia  of  a body , rotating  about  a given  axis, 
is  the  sum  of  the  products  obtained  by  multiplying  the  weights  of 
the  elementary  particles  of  which  it  is  composed  by  the  square  of 
their  distances  from  the  axis. 

It  is  often  desirable  to  use  the  moment  of  inertia  for  a plane 
section;  but  as  a plane  surface  has  no  weight,  it  is  apparent 
that  the  above  definition  does  not  correctly  apply.  The  fol- 
lowing definition  applies  to  plane  surfaces: 

The  moment  of  inertia  of  a plane  surface  about  a given  axis  is 
the  sum  of  the  products  obtained  by  multiplying  each  elementary 
areas  into  which  the  surface  may  be  conceived  to  be  divided  by 
the  square  of  its  distance  from  the  axis. 

The  axis  about  which  the  body  or  surface  rotates,  or  is 
assumed  to  rotate,  i.  e.,  the  axis  from  which  the  distance  to 
each  area  or  particle  is  measured,  is  called  the  axis  of  rotation. 
The  least  moment  of  inertia  is  that  value  of  the  moment  of 
inertia  of  a body  or  section  when  the  axis  of  rotation  passes 
through  the  center  of  gravity,  since  its  value  is  less  for  that 
position  of  the  axis  than  for  any  other. 

’ To  find  the  moment  of  inertia  of  a body  about  a given  axis: 

Divide  the  body  or  section  into  many  small  parts  and  multiply 
the  weight  or  area  of  each  part  by  the  square  of  the  distance  from 
its  center  of  gravity  to  the  axis  of  rotation;  the  sum  of  these 
products  will  be  the  moment  of  inertia. 

Note. — The  results  obtained  by  the  above  rules  are  really 
only  approximate;  for  practically  it  is  impossible  to  divide 
a body  or  surface  into  parts  sufficiently  small  for  absolute 
accuracy.  The  smaller  the  parts  the  more  accurate  will  be 
the  result;  but  the  results  obtained  by  these  rules  will  always 
be  slightly  too  small. 

The  moment  of  inertia  is  usually  designated  by  the  letter  I. 

Formulas  for  the  values  of  I about  an  axis  of  rotation 
passing  through  the  center  of  gravity  of  the  section  are  given 
for  various  forms  of  sections  in  Table  V,  page  153. 


126 


MECHANICS. 


The  moment  of  inertia  about  an  axis  of  rotation  not 
passing  through  the  center  of  gravity  is  equal  to  the  moment  of 
inertia  about  a parallel  axis  through . the  center  of  gravity  plus 
the  product  of  the  entire  weight  of  the  body  (or  area  of  the  section) 

' multiplied  by  the  square  of  the  distance  between  the  two  axes. 

Example.— It  is  desired  to  find  the  moment  of  inertia  of  a 
6"  I-beam  of  the  dimensions  shown  in  Fig.  1 about  an  axis 
x y perpendicular  to  the  web  of  the  beam  at  the  center. 

Solution.— Since  the  axis  about  which  the  moment  of 
inertia  is  to  be  found  is  an  axis  of  sym- 
metry of  the  beam,  it  is  necessary  to 
make  the  computations  only  for  the 
half  section  of  the  beam  lying  at  one 
side  of  the  axis,  and  multiply  the 
result  by  2.  As  stated  before,  the 
smaller  the  parts  into  which  the  area 
is  divided,  the  more  accurate  will  be 
the  result. 

It  wTill  be  sufficiently  accurate  for 
present  purposes  to  divide  the  section 
in  the  manner  shown  in  Fig.  2. 

The  operations  are  given  at  the  side 
of  the  figure,  and  will  be  readily  under- 
stood. The  sum  of  the  products  is  the  approximate  value  of 
the  moment  of  inertia  of  this  half  of  the  section  about  the 
axis  xy,  and  when  multiplied  by  2 is  the  approximate  value 


of  I for  the  entire  section. 

It  is  found  to  equal  23.444. 

Square  of 

Area. 

Distance. 

3.50  X 

.25  = .875 

.875  X 2.8752  = 7.232 

3.27  X 

.125  — .409 

.409  X 2.6672  = 2.907 

.23  X 

.50  = .115 

.115  X 2.502  = .719 

.23  X 

.50  = .115 

.115  X 2.002  = >460 

.23  X 

.50  = .115 

.115  X 1.502  = .259 

.23  X 

.50  = .115 

.115  X 1.002  = .115 

.23  X 

.50  = .115 

.115  X 0.502  = .029 

.23  X 

.25  = .058 

.058  X 0.1252  = .001 

1.917 

11.722 

2 

2 

A = 3.834 

I = 23.444 

CENTER  OF  OSCILLATION. 


127 


If  the  web  of  the  beam  is  divided  into  areas  1 in.  in 
height  (instead  of  £ in.),  the  value  of  I obtained  will  be 
23.46  in.  If  the  section  is  considered  to  be  of  the  form  indi- 
cated by  the  dotted  lines  in  Fig.  1,  and  to  have  the  same  area 
as  the  original  section,  then,  by  the  formula  for  the  moment  of 
inertia  of  an  I-beam  given  in  Table  V,  page  153,  the  value  of 

r 3.50  X 6s  — 3.27  X 5.253  c„ 

I - - = 23.57. 


The  true  value  is  almost  exactly  23.48  in.  Any  one  of 
these  values  would  be 
sufficiently  correct  for 
most  practical  pur- 
poses. 

If  it  is  desired  to  find  * 
the  moment  of  inertia  * 
of  a body  about  a given  * 
axis  with  reference  to  * 
the  weight  of  the  body,  * 
the  process  is  substan-  * 
tially  the  same  as  in 
the  example  given  for 
the  plane  section,  ex-  OLU 1 1 1 
cept  that  the  weight  of 

each  small  part  of  the  Fig.  2. 

body  is  taken  instead 

of  the  area  of  each  small  part  of  the  section. 


CENTER  OF  OSCILLATION. 

The  center  of  oscillation  of  a pendulum  or  other  body 
vibrating  or  rotating  about  a fixed  axis  or  center  is  that 
point  at  which,  if  the  entire  weight  of  the  body  were  con- 
centrated, the  body  would  continue  to  vibrate  in  the  same 
intervals  of  time. 

When  a pendulum,  or  other  suspended  body,  is  oscillating 
backward  and  forward,  it  is  plain  that  those  particles  that 
are  farther  from  the  point  of  suspension  travel  through 
greater  distances,  and  therefore  move  with  greater  velocities 
than  those  particles  that  are  nearer  the  point  of  suspension. 


128 


MECHANICS. 


But  there  is  evidently  some  point  on  the  pendulum  that 
travels  through  the  same  distance  and  has  the  same  velocity 
as  the  average  distance  and  average  velocity  of  all  the  par- 
ticles. This  point  is  called  the  center  of  oscillation;  it  is  not 
situated  at  the  center  of  gravity.  It  always  exists  in  the  ball 
of  a revolving  governor  or  other  rotating  body.  The  axis  or 
center  around  which  the  body  rotates  (corresponding  to  the 
point  of  suspension  in  pendulum)  is  the  axis  of  rotation. 

The  distance  from  the  axis,  or  center  of  rotation,  to  the 
center  of  oscillation  is  sometimes  called  the  true  length  of  the 
pendulum;  it  is  also  called  the  radius  of  oscillation;  the  latter 
name  is  preferable.  To  find  the  radius  of  oscillation: 

Divide  the  moment  of  inertia  of  the  body  about  the  given  axis 
of  rotation  by  the  product  of  the  total  weight  of  the  body , multi- 
plied by  the  distance  from  the  given  axis  to  the  center  of  gravity 
of  the  body. 

The  centers  of  oscillation  and  of  rotation  (point  of  suspen- 
sion) are  interchangeable.  If  the  position  of  a pendulum  is 
reversed,  and  suspended  from  its  center  of  oscillation,  the 
pendulum  will  vibrate  in  the  same  intervals  of  time. 

Example. — It  is  desired  to  find  the  position  of  the  center 
of  oscillation  of  a wrought-iron  bar  1 in.  square  and  12  in. 
long,  axis  of  rotation  perpendicular  to  the  bar  at  one  end: 


Weight 
of  Each 
Cu.  In. 

Sq.of 

Dist. 

.281 

X 

p 

Cn 

II 

0.070 

.281 

X 

1.52  = 

0.632 

.281 

X 

2.52  = 

1.756 

.281 

X 

3.52  = 

3.442 

.281 

X 

II 

fh 

5.690 

.281 

X 

II 

?c 

10 

8.500 

.281 

X 

6.52  = 

11.872 

.281 

X 

7.52  = 

15.806 

.281 

X 

8.52  = 

20.302 

.281 

X 

9.52  = 

25.360 

.281 

x: 

10.52  = 

30.980 

.281 

x: 

Ll.52  = 

37.162 

-M- 

i-m-jr- 

Jjj 

ira 

--- 

L_U_ 

u? 

... 

|+b 

0: 

— 

[1 

-- 

j * 

7U 

.JO  -II 

J-S-5- 

2-~ 


TF 


3.372 


161.572  = I 


CENTER  OF  PERCUSSION. 


129 


Solution.— For  the  purposes  of  the  example  it  will  be 
sufficiently  accurate  to  find  the  moment  of  inertia  by  con- 
sidering the  bar  to  be  divided  into  12  equal  cubes,  each  con- 
taining 1 cu.  in.  of  metal,  as  indicated  in  the  figure,  and  the 
weight  of  each  cube  to  be  concentrated  at  its  center  of  gravity. 

The  weight  of  1 cu.  in.  of  wrought  iron  is  .281  lb.,  and  of  a 
bar  1 in.  square  and  1 ft.  long  it  is  .281  X 12  = 3.372  lb. 
Hence,  I = .281  X .52  + .281  X 1.52  + etc.  = 161.572.  (See 
page  128.)  The  exact  value  of  I is  161.856;  this  shows  that 
the  approximate  method  is  very  close. 

According  to  the  rule  previously  given,  if  the  moment  of 
inertia  is  divided  by  the  product  of  the  weight  of  the  body, 
by  the  distance  from  the  axis  of  rotation  to  the  center  of 
gravity,  the  quotient  will  be  the  radius  of  oscillation. 

Therefore,  the  distance  from  the  exact  center  of  oscillation 
of  a wrought-iron  bar,  1 in.  square  and  12  in.  long,  to  an  axis 
of  rotation  perpendicular  to  the  end  of  the  bar,  is 


161.856  . . 

3.372  X 6 8m'’ 

or  two-thirds  of  the  length  of  the  bar. 

The  value  of  I for  a bar  of  any  cross-section,  provided  it  is 
uniform  throughout  its  length,  revolving  about  an  axis  per- 
pendicular to  it  and  passing  through  its  end,  is 


WP 
3 ’ 


in  which  W is  the  weight  of  the  bar,  and  l is  its  length. 


Hence, 


I = 


WV  3.372  X 122 


= 161.856. 


3 3 

If  the  axis  passes  through  the  center  of  gravity  of  the  bar, 


I 


WV 
12  ’ 


CENTER  OF  PERCUSSION. 

The  center  of  percussion  with  respect  to  a given  axis  of 
rotation  may  be  defined  as  the  point  of  application  of  the 
resultant  of  the  forces  that  cause  the  body  to  rotate.  It  is 
that  point  at  which  if  a force  is  applied,  the  force  will  have 
no  effect  at  the  axis  of  rotation. 


130 


MECHANICS. 


Strike  anything  solid,  as  an  anvil,  with  a stick.  If  the  ' 
end  of  the  stick  hits  the  anvil,  the  opposite  end  will  sting 
your  hand  and  will  jerk  in  the  direction  in  which  the  blow  is 
struck;  if  the  center  of  the  stick  hits  the  anvil  it  will  again 
sting  your  hand,  but  you  will  jerk  it  in  a direction  opposite 
to  the  movement  of  the  blow.  But  somewhere  between  the 
end  and  the  center  of  the  stick  will  be  a point  where  it  may 
hit  the  anvil  and  not  sting  your  hand  at  all.  This  point  is 
the  center  of  percussion. 

Level  off  the  surface  of  some  wet  sand  and  lay  a strip  of 
board  upon  it  (say  18  in.  long  and  3 in.  wide).  Strike  or 
press  the  board  near  the  center  and  the  entire  length  of  the 
board  will  be  imprinted  in  the  sand;  but  press  it  near  one 
end  and  the  opposite  end  will  be  raised  up  from  the  sand 
and  will  make  no  imprint.  Between  the  center  and  the  end 
of  the  board  is  a point  that  if  pressed  upon  will  cause  no 
movement  in  the  opposite  end,  i.  e.,  the  end  of  the  board 
will  neither  press  into  the  sand  nor  be  lifted  from  it,  but  the 
imprint  in  the  sand  will  diminish  to  zero  at  the  end  of 
the  board.  The  point  pressed  or  struck  will  be  the  center 
of  percussion.  If  the  board  is  of  uniform  width,  the  center 
of  percussion  will  be  at  one-third  of  the  distance  from  one 
end  of  the  board. 

Similarly  in  the  preceding  illustration,  if  the  stick  is  of  uni- 
form size  and  weight,  and  your  hand  grasps  it  at  one  end,  the 
point  at  which  it  can  strike  the  anvil  without  affecting  your 
hand  will  be  at  one-third  the  distance  from  the  opposite  end. 

In  all  cases  the  center  of  percussion  is  identical  with  the  center 
of  oscillation , and  its  position  is  found  in  the  same  manner. 

Example.— It  is  desired  to  find  the  position  of 
the  center  of  oscillation  or  percussion  of  two  balls 
fastened  upon  a rod.  The  first,  weighing  2 lb.,  is 
at  a distance  of  18  in.  from  the  axis  of  rotation, 
and  the  second,  weighing  1 lb.,  is  at  a distance  of 
36  in.  from  the  axis.  (See  figure.) 

Solution.— For  simplicity,  the  rod  will  be 
assumed  to  have  no  weight.  Consider  the  weight 
of  each  ball  to  be  concentrated  at  its  center  of 
gravity. 


RADIUS  OF  GYRATION. 


131 


The  moment  of  inertia  is  found  as  follows. 


Sq.  of 
Wt.  Dist. 

2 X 182  = 
1 X J362  = 


648 

1,296 


1,944  = I. 

The  center  of  gravity  of  the  two  balls  is  found  to  be  at  a 
distance  of  6 in.  from  the  larger,  or  24  in.  from  the  axis  of  rota- 
tion (see  page  124),  and  the  combined  weight  of  the  two  balls 
is  2 + 1 = 3 lb.  Therefore,  the  center  of  percussion  is  found 
1 944 

to  be  at  a distance  of  g ^ — = 27  in.  from  the  axis  of  rotation. 

But,  in  an  actual  case,  the  rod  would  have  weight,  and  its 
moment  of  inertia  must  be  considered  as  w'ell  as  the  moment 
of  inertia  of  the  balls. 

If  we  assume  that  the  rod  is  of  steel,  f in.  in  diameter  and 
86  in.  long,  it  will  weigh  X .7854  X 36  X -283  = 1.125  lb. 
.283  lb.  is  the  weight  of  1 cu.  in.  of  steel. 

Using  the  formula  given  on  page  129, 

I=JFg  = L125X3g  = 486- 

Adding  this  result  to  the  former,  1,944  + 486  = 2,430  = 
moment  of  inertia  of  rods  and  balls.  The  center  of  gravity 
of  the  combination  is  found  by  the  formula  (see  page  124) 

pP+w-  Substituting,  = 1TV  24  - lT7r  = 22x\  in. 

= distance  from  end  of  rod  to  center  of  gravity. 

Applying  the  rule  given  for  finding  the  center  of  oscilla- 
tion, the  distance  of  the  center  of  percussion  from  the  end  of 
2 430 

thebarls(1  + 2 + 1'125)x-22A  = 26.34 in.,  very  nearly. 


RADIUS  OF  GYRATION. 

The  center  of  gyration  is  that  point  in  a revolving  body  at 
which,  if  the  entire  mass  of  the  body  were  concentrated,  the 
moment  of  inertia  with  respect  to  a given  axis  would  be  the 
same  as  in  the  body. 

An  ounce  of  cork  occupies  about  94  times  as  much  space  as 


132 


MECHANICS. 


an  ounce  of  platinum;  but  the  ounce  of  platinum  can  have 
the  same  moment  of  inertia  as  the  ounce  of  cork,  if  its  center 
of  gyration  has  the  same  position  with  respect  to  the  axis  of 
rotation. 

The  center  of  gyration  is  not  at  the  center  of  gravity , nor  >at 
the  center  of  oscillation , but  at  some  point  in  a straight  line 
between  those  centers. 

The  radius  of  gyration  is  the  distance  from  the  axis  of 
rotation  to  the  center  of  gyration. 

The  square  of  the  radius  of  gyration  is  the  average  of  the 
squares  of  the  distances  from  the  axis  of  rotation  to  each  ele- 
mentary particle  of  the  body,  or  to  each  elementary  area  of  the 
section,  as  the  case  may  be.  But  the  sum  of  these  squares  of 
distances,  multiplied  by  the  weight  or  area  of  each  ele- 
mentary part,  equals  the  moment  of  inertia;  therefore,  the 
moment  of  inertia  divided  by  the  weight  of  the  body  or  area 
of  the  section  equals  the  square  of  the  radius  of  gyration;  the 
square  root  of  this  quotient  is  the  radius  of  gyration. 

But,  according  to  the  rule  for  finding  the  radius  of  oscil- 
lation, the  quotient  obtained  by  dividing  the  moment  of 
inertia  by  the  weight  or  area  equals  the  product  of  the  dis- 
tance from  the  axis  of  rotation  to  the  center  of  gravity,  mul- 
tiplied by  the  radius  of  oscillation;  and,  therefore,  the  radius 
of  gyration  is  a mean  proportional  between  these  distances. 

If  the  distance  from  the  axis  of  rotation  to  the  center  of 
gravity  is  known,  and  the  radius  of  oscillation  is  known,  the 
radius  of  gyration  may  be  found  by  multiplying  these  two 
known  distances  together  and  extracting  the  square  root  of 
the  product. 

In  the  example  of  the  I-beam,  Fig.  2,  page  126,  the  sum  of 
the  areas  of  the  half  section  of  the  beam  is  1.917,  and  the 
area  of  the  entire  section  is  3.834  sq.  in.  Therefore,  the  radius 
of  gyration  of  this  beam  about  an  axis  through  the  center  of 

/2344 

gravity  perpendicular  to  the  web  = = 2.47  in. 

In  the  example  of  the  iron  bar  12  in.  long  (see  figure, 
page  128),  the  distance  from  the  axis  of  rotation  to  the  center 
of  gravity  is  6 in.,  and  the  radius  of  oscillation  was  found  to 
equal  8 in.  Therefore,  the  radius  of  gyration  about  an 


RADIUS  OF  GYRATION. 


133 


axis  perpendicular  to  the  bar  at  one  end  = j/  6 X 8 = 6.93  in. 
Or,  the  moment  of  inertia  of  the  bar  = 161.586,  and  the 
weight  of  the  bar  = 3.372  lb.  Therefore,  the  radius  of  gyra- 


tion 


,'161.586 
\ 3.372 


6.93  in.,  very  nearly. 


The  radius  of  gyration  is  used  in  determining  the  strength 
of  columns.  The  axis  must  be  taken  in  such  a direction  that 
the  result  will  be  the  least  radius  of  gyration  of  the  column; 
this  condition  is  usually  obtained  when  the  axis  is  perpen- 
dicular to  the  least  diameter  or  side  of  the  column. 

The  various  relations  between  these  quantities  may  be 
concisely  expressed  by  the  following  formulas,  in  which 
A = area  of  section  (or  weight  of  body  if  the  weight  is  used); 
g = distance  from  axis  of  rotation  to  center  of  gravity; 

G = radius  of  gyration; 
r0  = radius  of  oscillation; 

I = moment  of  inertia. 

Then, 


I — A G2. 

I = Agr0. 

% 

II 

/-p 

I 

I 

G = \^‘ 

9 ~ Ar0' 
G2 

c"S 

II 

G = v gr0. 

9 = 7- 
' 0 

r0  = - — . 

y 

g:G  = G:  ra. 


To  find  the  radius  of  oscillation,  radius  of  gyration,  and 
moment  of  inertia,  experimentally. 

The  connecting-rod  of  an  engine  is  represented  in  the 


figure.  It  is  desired  to  find  the  moment  of  inertia  of  the  rod 
about  an  axis  of  rotation  through  the  center  of  the  crosshead 
pin  A. 

This  may  l)e  accomplished,  experimentally,  as  follow’s: 
Suspend  the  rod  from  the  crosshead  pin  in  such  a manner 


134 


MECHANICS. 


that  it  will  swing  freely;  cause  it  to  swing,  or  oscillate,  and 
note  the  exact  time  of  the  vibrations.  Remove  the  crosshead 
pin  and  reverse  the  rod,  but,  instead  of  suspending  it  by  the 
crankpin,  suspend  it  by  a movable  pin  Bf  that  can  be  clamped 
at  any  desired  point  upon  the  rod.  C is  another  view  of  this 
pin.  There  will  be  a point  on  the  rod  from  which  it  may  be 
suspended  by  means  of  the  movable  pin,  so  that  it  will  vibrate 
in  exactly  the  same  intervals  of  time  as  when  suspended  from 
the  crosshead  pin.  This  point  is  the  center  of  oscillation , for 
the  center  of  oscillation  and  the  center  of  rotation  are  inter- 
changeable; the  point  will  be  found  at  about  one-third  the 
length  of  the  rod  from  the  crankpin.  Find  this  center  of 
oscillation,  experimentally,  and  carefully  measure  the  dis- 
tance from  the  center  of  the  movable  pin  to  the  center  of  the 
crosshead-pin  hole.  This  distance  is  the  radius  of  oscillation 
— r0.  Next  remove  the  movable  pin,  and  find  the  center  of 
gravity  (lengthwise)  of  the  rod  by  balancing  it  across  a knife 
edge,  and  measure  the  distance  from  the  center  of  gravity 
thus  found  to  the  center  of  the  crosshead-pin  hole;  this  dis- 
tance = g.  Finally,  weigh  the  rod. 

The  product  of  the  weight  (=  A),  the  radius  of  oscillation 
(==  r0),  and  the  distance  from  the  center  of  crosshead  pin  (axis 
of  rotation)  to  the  center  of  gravity  (=  g)  will  be  the  moment 
of  inertia.  For,  by  the  formula,  I = A g r0.  The  radius  of 
gyration  G may  be  found  by  the  formula 


G = + or  G = j/ gr 

Ma 


MOMENT  OF  RESISTANCE. 

If  the  moment  of  inertia  of  the  cross-section  of  a beam  is 
divided  by  the  distance  from  the  neutral  axis  (see  definition 
on  next  page)  to  the  extreme  fiber,  i.  e.,  the  fiber  that  is  far- 
thest from  the  axis,  the  quotient  will  be  the  quantity  known 
as  the  moment  of  resistance. 

It  is  evident  that,  if  a beam  is  strained  by  a vertical  load, 
the  greatest  stress  will  be  in  the  extreme  upper  and  lower 
fibers  of  the  beam. 


MOMENT  OF  RESISTANCE. 


135 


The  intensity  of  the  stress  that  can  be  borne  by  the  extreme 
fibers  is  the  limit  of  the  strength  of  the  beam. 

The  upper  fibers  are  compressed  and  the  lower  fibers  are 
stretched,  but  somewhere  along  or  near  the  center  of  a 
vertical  section  of  the  beam,  the  fibers  are  neither  extended 
nor  compressed;  the  position  of  these  fibers  is  called  the 
neutral  surface , and  the  line  where  this  neutral  surface  inter- 
sects a right  section  of  the  beam  is  the  neutral  axis  of  the 
section. 

The  neutral  axis  passes  through  the  center  of  gravity  of 
the  section. 

If  the  moment  of  resistance  is  multiplied  by  the  amount  of 
stress  that  may  be  allowed  per  square  inch  upon  the  extreme 
fiber,  the  product  will  represent  the  efficiency  of  the  beam 
to  resist  bending  moment. 

Example.— Referring  to  the  6"  I-beam,  Figs.  1 and  2, 
pages  126  and  127,  for  which  the  moment  of  inertia  of  the 
section  has  been  found,  it  is  desired  to  ascertain  the  load  that 
a wrrought-iron  beam  of  the  same  dimensions  as  Fig.  1 will 
carry  at  the  center  of  a span  8 ft.  between  supports. 

Solution.— The  moment  of  resistance  for  the  section  = 
23  48 

— = 7.83.  In  Table  II,  page  151,  the  ultimate  strength  or 

fiber  stress  for  wrought  iron  is  given  as  50,000  lb.  per  sq.  in., 
and  in  Table  I,  page  151,  the  factor  of  safety  given  for 
wrought  iron  under  a steady  stress  is  4;  therefore,  the  safe 

fiber  stress  for  wrought  iron  = ~ = 50,^-Q-  = 12,500  lb.  per 
sq.  in.,  and  the  moment  of  resistance  multiplied  by  the  safe 

CJ  D 

fiber  stress,  or  — = 7.83  X 12,500  = 97,875  in.-lb.  But  l = 
8 ft.,  or  96  in.;  equating  the  bending  moment  for  a load  at 


the  center  of  a beam 


(- 


Wl\ 

4 > 


with  the  moment  of  resist- 


_ __  SR  Wl  96  W .. 

ance,  or  putting  M = — — = — — ; then  — — = 97,875;  there- 
4 4 4 


fore,  W = 4,078  lb.,  the  load  that  can  be  safely  supported  at  the 
center  of  the  beam. 


136 


MECHANICS. 


MECHANICAL  POWERS. 


F : W = l : L.  FL  = Wl. 


F = 


Wl 


w = 


FL 


Fa 


F = 


W — 


F—  W' 

> 

' 

II 

II 

& 

.F12  = TFr. 

Wr 

TFr 
E ~ ~F~' 

R * 

RF 

RF 

r 

r = ^F* 

F = 


Wrr ' 
~RR'  ' 


IF  = 


.F1212' 


n = number  of  revolutions  of  large  gear. 
n : n'  = r' : 22. 
v : vf  = rr':R  12'. 

v = velocity  of  TF;  v'  = velocity  of  .F. 


TFrr'r"  FRR'R" 

~RR'R'r  W ~ rr'r'-  * 
n:n"  = r'  r"  : 1222'. 
v:v'  = r r'  r"  : 12 12'  12". 
r,  r',  r",  etc.  = radii  of  the  pinions; 
R,  22',  12",  etc.  = radii  of  the  wheels. 


MECHANICAL  POWERS. 


137 


Let  db  and  qb  represent  the  magnitudes  and  direc- 
tions of  two  forces  that  act  to  move  the 
body  b.  By  completing  the  parallel- 
ogram there  will  be  obtained  a diagonal 
force  fb , whose  magnitude  and  direction 
are  equal  to  the  effect  produced  by  d b and 
q b.  fb  is  called  the  resultant  of  d b and  q b. 

If  three  or  more  forces  act  in  different  directions  to 
move  a body  b,  find  the  resultant  of 
any  two  of  them,  and  consider  it  as 
a single  force.  Between  this  and  the 
next  force  find  a second  resultant.  Thus, 
pb,  qb,  and  r b are  magnitudes  and 
directions  of  the  forces,  pb  qb  + rb  = 
gb  + rb  = fb,  the  magnitude  and  direc- 
tion of  the  three  forces,  pb,  qb,  and  r b. 


A SINGLE  riXED  PULLEY. 

F = W. 

V = Vf. 

v = velocity  of  IE;  v'  — velocity  of  F. 


A SINGLE  MOVABLE  PULLEY. 

F : W = 1 : 2,  or  F = £ W. 

If  the  force  F be  applied  at  a and  act 
upwards,  the  result  will  be  the  same. 
v'  = 2 v. 

v = velocity  of  W ; v'  = velocity  of  F. 


A DOUBLE  MOVABLE  PULLEY,, 

F : W = 1 : 4,  or  F = £ W. 

Let  u = number  of  parts  of  rope,  not 
counting  the  free  end. 

F = W -T-  u.  v : v'  = 1 : u. 
v = velocity  of  W ; v'  — velocity  of  F. 


138 


MECHANICS. 


QUADRUPLE  MOVABLE  PULLEY. 

F = I W.  F : W = 1 : 8. 

Let  u = number  of  parts  of  rope,  not 
counting  the  free  end;  then, 

F — W H-  u.  v : v'  = 1 : u. 
v = velocity  of  W;  v'  = velocity  of  F. 


COMPOUND  PULLEY. 

u = number  of  movable  pulleys. 

W 

F = ~ W = 2U  F. 

2“ 

v : vf  = 1 : 2“. 

v = velocity  of  IT;  v'  = velocity  of  F. 


DIFFERENTIAL  PULLEY. 

2 PE 


W = 


E-r’ 


AN  OBLIQUE  FIXED  PULLEY. 

F : IF  = 1 : 2 cos  z. 

w 


INCLINED  PLANE. 


MECHANICAL  POWERS. 


139 


SCREW. 

P = pitch  of  the  screw; 
r = radius  on  which  the  force  jPacts. 
F : W : : P:  2 nr. 


F = 


WP 
2 n r 


W = 


2nr  F 


Work  is  the  overcoming  of  resistance  through  a distance. 
The  unit  of  work  is  the  foot-pound;  that  is,  it  equals  1 pound 
raised  vertically  1 foot.  The  amount  of  work  done  is  equal  to 
the  resistance  in  pounds  multiplied  by  the  distance  in  feet 
through  which  it  is  overcome.  If  a body  is  lifted,  the  resist- 
ance is  the  weight  or  the  overcoming  of  the  attraction  of 
gravity,  the  work  done  being  the  weight  in  pounds  multiplied 
by  the  height  of  the  lift  in  feet.  If  a body  moves  in  a hori- 
zontal direction,  the  work  done  is  the  friction  overcome,  or 
the  force  needed  to  move  a resistant  body  or  combination  of 
bodies,  multiplied  by  the  distance  moved.  In  order  to  com- 
pare the  different  amounts  of  work  done  by  different  systems 
of  forces,  time  is  also  considered. 

One  horsepower  is  550  ft.-lb.  of  work  in  1 second,  or  33,000 
ft.-lb.  in  1 minute,  or  1,980,000  ft.-lb.  in  1 hour. 

The  work  necessary  to  be  done  in  raising  a body  weighing 
W lb.  through  a height  of  h ft.  equals  Wh  ft.-lb.  The  total 
work  that  any  moving  body  is  capable  of  doing  in  being 

Wv* 

brought  to  rest  equals  its  kinetic  energy,  or  — — , when  v is 

2 g 

the  velocity  in  feet  per  second. 

Thus,  the  work  that  a cannon  ball  weighing  800  lb.  and 
traveling  with  a velocity  of  1,200  ft.  per  sec.  could  do,  is 

If  stopped  in  1 min.,  the  horsepower  would  be  17,910,447 
-s-  33,000  = 542.8,  nearly. 


140 


MECHANICS. 


FORCE  OF  A BLOW. 

In  order  to  determine  the  force  of  a blow,  the  velocity  of 
the  object  at  the  instant  of  striking  must  be  known,  and  also 
the  time  required  to  bring  the  body  to  rest.  It  is  a very 
difficult  matter  to  determine  the  exact  time,  but  a close 
approximation  to  the  striking  force  may  be  obtained  by 
dividing  the  kinetic  energy  of  the  body  at  the  instant  of  stri- 
king by  the  average  amount  of  penetration  or  compression 
produced  by  the  striking  body. 

Let  F — striking  force  in  pounds; 

W = weight  of  striking  body  in  pounds; 
v = velocity  of  striking  body  in  feet  per  second; 

R — distance  penetrated  or  amount  of  compres- 
sion = the  distance  through  which  the  resist- 
ance acts,  in  feet; 

t — time  required  to  bring  the  body  to  rest; 
h = height  in  feet  which  would  produce  the  veloc- 
ity v. 

rm,  ^ WV  _ WV2  Wh 

gt  2 g R R 

Example.— A steam  hammer  weighing  1,000  lb.  (with  its 
piston)  falls  from  a height  of  8 ft.,  and  compresses  a piece  of 
iron  £ in.;  what  is  its  striking  force? 

Solution.— If  gravity  be  considered  as  the  only  force 
acting,  the  steam  on  top  of  the  piston  being  used  to  prevent 
a rebound  of  the  hammer, 

„ Wh  1,000  X 8 

f=-r-  = — 


■ = 1,000  x 8 x 8 x 12  = 768,000  lb. 


(*  + 12) 

Divide  | in.  by  12,  to  obtain  the  amount  of  compression 
in  feet  or  parts  of  a foot. 


BELTING. 

D = diameter  of  larger  pulley  in  inches; 
d = diameter  of  smaller  pulley  in  inches; 

N = revolutions  per  minute  of  larger  pulley; 
n = revolutions  per  minute  of  smaller  pulley; 

W = width  of  double  belt  in  inches; 
w = width  of  single  belt  in  inches; 

H = horsepower  that  can  be  transmitted  by  the  belt. 


BELTING. 


141 


Then,  H = — for  single  belts. 

2,750 

I)  N W 

II  = , for  double  belts. 

1,925 

2,750  H 2,750.  H 
w ~~  j)jsf  — dn 
1,925  H 1,925  H 
W~  DN  ~ dn  ' 

D = for  single  belt. 

wN 

D = for  double  belt. 

N = 2,/50  ^ for  single  belt. 
wD 

2VT  = for  double  belt. 

W D 

The  above  rules  are  for  open  belts  and  pulleys  having  the 
same  diameter,  the  arc  of  contact  being,  in  this  case,  half  the 
circumference,  or  180°.  For  open  belts  and  pulleys  of  different 
diameters,  the  arc  of  contact  is  less  than  180°  on  the  smaller 
pulley,  and  a different  constant,  to  be  taken  from  the  fol- 
lowing table,  must  be  substituted  in  the  formulas.  To  find 
the  arc  of  contact,  let  l be  the  distance  in  inches  between  the 

centers  of  the  pulleys.  Then,  = cosine  of  half  the  angle 

Find  this  half  angle  from  a table  of  natural  cosines,  and 


Degrees. 

Fraction  of 
Circumference. 

Single  Belt 
Constant. 

Double  Belt 
Constant. 

90 

V±  = .25 

6,080 

4,250 

11234 

A - .3125 

4,730 

3,310 

120 

y = .3333 

4,400 

3,080 

135 

% = .375 

3,850 

2,700 

150 

A = .4167 

3,410 

2,390 

15734 

* = .4375 

3,220 

2,250 

180  to  270 

34  to  % = .5  to  .75 

2,750 

1,925 

multiply  by  2.  The  result  is  the  arc  of  contact  in  degrees. 
Find  the  number  in  the  first  column  of  the  table,  wThich  is 
nearest  to  this  result,  and  use  the  constant  corresponding  to 


142 


MECHANICS. 


that  number.  If  a table  of  natural  cosines  is  not  at  hand, 
measure  the  length  of  the  arc  of  contact  on  the  smaller  pulley 
and  divide  it  by  the  circumference  of  the  pulley.  Find  the 
fraction  in  the  Second  column  that  corresponds  nearest  to 
this  result,  and  opposite  this  its  corresponding  constant. 

Example.— What  must  be  the  width  of  a single  belt  to 
transmit  12  horsepower,  when  the  diameter  of  the  larger 
pulley  is  42  in.,  of  the  smaller  pulley  20  in.,  distance  between 
their  centers  14  ft.  = 168  in.,  and  R.P.M.of  smaller  pulley  150? 

42 20 

Solution.—  2 ^ — = .06548  = cosine  of  half  the  arc  of 
contact,  which  thus  = 86°  15',  nearly;  86°  15' X 2 = 172£°  = 
arc  of  contact;  the  nearest  number  in  the  table  is  180°,  and  the 

corresponding  constant  is  2,750;  hence,  w = = 11  in. 

Oak-tanned  leather  makes  the  best  belts.  When  belts  are 
run  with  the  hair  side  over  the  pulley,  they  have  greater 
adhesion. 

The  ordinary  thickness  of  leather  belts  is  ^ in.,  and 
their  weight  is  about  60  lb.  per  cu.  ft. 

Ordinarily,  four-ply  cotton  belting  is  considered  equivalent 
to  single-leather  belting. 


RULES  FOR  CALCULATING  THE  SPEED  OF  GEARS 
OR  PULLEYS. 

In  calculating  for  gears,  multiply  or  divide  by  the  diameter 
or  the  number  of  teeth,  as  may  be  required.  In  calculating 
for  pulleys,  multiply  or  divide  by  their  diameters  in  inches. 

The  driving  wheel  is  called  the  driver , and  the  driven 
wheel  the  driven  or  follower. 

Problem  I. 

The  revolutions  of  driver  and  driven,  and  the  diameter  of 
the  driven,  being  given,  required  the  diameter  of  the  driver. 

Rule.— Multiply  the  diameter  of  the  driven  by  its  number  of 
revolutions , and  divide  by  the  number  of  revolutions  of  the  driver. 

Problem  II. 

The  diameter  and  revolutions  of  the  driver  being  given, 
required  the  diameter  of  the  driven  to  make  a given  number 
of  revolutions  in  the  same  time. 


PUMPS. 


143 


Rule. — Multiply  the  diameter  of  the  driver  by  its  number  of 
revolutions,  and  divide  the  product  by  the  required  number  of 
revolutions. 

Problem  III. 

The  diameter  or  number  of  teeth,  and  number  of  revolu- 
tions of  the  driver,  with  the  diameter  or  number  of  teeth  of 
the  driven,  being  given,  required  the  revolutions  of  the 
driven. 

Rule. — Multiply  the  diameter  or  number  of  teeth  of  the  driver 
by  its  number  of  revolutions , and  divide  by  the  diameter  or  num- 
ber of  teeth  of  the  driven. 

Problem  IV. 

The  diameter  of  driver  and  drwen,  and  the  number  of 
revolutions  of  the  driven,  being  given,  required  the  number 
of  revolutions  of  the  driver. 

Rule. — Multiply  the  diameter  of  the  driven  by  its  number  of 
revolutions,  and  divide  by  the  diameter  of  the  driver. 


PUMPS. 

In  all  pumps,  whether  lifting,  force,  steam,  single-acting, 
double-acting,  or  centrifugal,  the  number  of  foot-pounds  of 
work  performed  by  the  pump  is  equal  to  the  weight  of  the 
water  discharged  in  pounds,  multiplied  by  the  vertical  dis- 
tance in  feet  between  the  level  of  the  water  in  the  well  or 
source  and  the  point  of  discharge,  plus  the  work  done  in 
overcoming  the  friction  and  other  resistances.  (It  is  assumed 
that  the  water  is  delivered  with  practically  no  velocity.) 

To  find  the  discharge  of  a pump  in  gallons  per  minute: 

Let  T = piston  travel  in  feet  per  minute; 
d = diameter  of  cylinder  in  inches; 

G = number  of  gallons  discharged  per  minute. 

Then,  G = .03264  Td *. 

To  find  the  horsepower  of  a pump,  use  the  following 
formula,  in  which  Tand  d are  the  same  as  above,  and  h is  the 
vertical  distance  in  feet  between  the  level  of  the  water  at 
the  source  and  the  point  of  discharge: 

H.  P.  = .00033724  Gh  = .00001238  Td*  h. 

Jn  both  the  above  formulas,  allowance  has  been  made  ior 
friction,  leakage,  etc. 


144 


MECHANICS. 


DUTY. 

The  duty  of  a pump  is  the  number  of  foot-pounds  of  work 
actually  done  for  100  lb.  of  coal  burned. 

Duty  = 835.53 

w 

where  W = weight  of  coal  burned,  in  pounds. 


HYDROMECHANICS. 


HYDROSTATICS. 

Hydrostatics  treats  of  liquids  at  rest  under  the  action  of 
forces.  If  a liquid  is  acted  on  by  a pressure,  the  pressure  per 
unit  of  area  exerted  anywhere  on  the  mass  of  liquid  is  trans- 
mitted undiminished  in  all  directions,  and  acts  with  the 
same  force  on  all  surfaces,  in  a direction  at  right  angles  to 
those  surfaces. 

General  Law  for  the  Downward  Pressure  on  the  Bottom  of  Any 
Vessel.— The  pressure  on  the  bottom  of  a vessel  containing  a 
liquid  is  independent  of  the  shape  of  the  vessel,  and  is  equal 
to  the  weight  of  a prism  of  the  liquid  whose  base  is  the  same 
as  the  bottom  of  the  vessel,  and  whose  altitude  is  the  distance 
between  the  bottom  and  the  upper  surface  of  the  liquid,  plus 
the  pressure  per  unit  of  area  upon  the  upper  surface  of  the 
liquid  multiplied  by  the  area  of  the  bottom  of  the  vessel. 

General  Law  for  Upward  Pressure.— The  upward  pressure  on 
any  submerged  horizontal  surface  equals  the  weight  of  a 
prism  of  the  liquid  whose  base  has  an  area  equal  to  the  area  of 
the  submerged  surface,  and  whose  altitude  is  the  distance 
between  the  submerged  surface  and  the  upper  surface  of  the 
liquid,  plus  the  pressure  per  unit  of  area  on  the  upper  surface 
of  the  liquid  multiplied  by  the  area  of  the  submerged  surface. 

General  Law  for  Lateral  P ressu re. -^The  pressure  on  any  ver- 
tical surface  due  to  the  weight  of  the  liquid  is  equal  to  the 
weight  of  a prism  of  the  liquid  whose  base  has  the  same  area 
as  the  vertical  surface,  and  whose  altitude  is  the  depth  of  the 
center  of  gravity  of  the  vertical  surface  below  the  level  of 
the  liquid.  Any  additional  pressure  is  to  be  added,  as4n  the 
previous  cases. 


HYDROMECHANICS. 


145 


Pressure  on  Oblique  Surfaces.— The  pressure  exerted  by  a 
liquid  in  any  direction  on  a plane  surface  is  equal  to  the 
weight  of  a prism  of  the  liquid  whose  base  is  the  projection 
of  the  surface  at  right  angles  to  the  given  direction,  and 
whose  height  is  the  depth  of  the  center  of  gravity  of  the 
surface  below  the  level  of  the  liquid. 

If  a cylinder  is  filled  with  water,  and  a pressure  applied, 
the  total  pressure  on  any  half  section  of  the  cylinder  is  equal 
to  the  projected  area  of  the  half  cylinder  (or  the  diameter 
multiplied  by  the  length  of  the  cylinder)  multiplied  by  the 
depth  of  the  center  of  gravity  of  the  half  cylinder,  multiplied 
by  the  weight  of  a cubic  inch  of  water,  plus  the  diameter  of 
the  shell,  multiplied  by  the  pressure  per  square  inch,  multi- 
plied by  the  length  of  the  cylinder. 

If  d = the  diameter,  and  l = the  length  of  the  cylinder, 
the  pressure  due  to  the  weight  of  the  water  when  the 

cylinder  is  vertical  upon  the  half  cylinder  = d X l X ^ X the 

1 2 

weight  of  a cubic  inch  of  water  = d X X the  weight  of  a 
cubic  inch  of  water;  d and  l are  to  be  measured  in  inches. 

The  pressure  in  pounds  per  square  inch  due  to  a head  of 
water  is  equal  to  the  head  in  feet  multiplied  by  .434. 

The  head  equals  the  pressure  in  pounds  per  square  inch 
multiplied  by  2.304. 

Example.— (a)  What  is  the  pressure  per  square  inch  cor- 
responding to  a head  of  water  of  175  ft.  ? (6)  If  the  pressure 
had  been  90  lb.  per  sq.  in.,  what  would  the  head  have  been? 

Solution.— (a)  175  x .434  = 75.95  lb.  per  sq.  in. 

(5)  90  X 2.304  = 207.36  ft. 


HYDROKINETICS. 

Hydrokinetics , also  called  hydrodynamics  and  hydraulics, 
treats  of  water  in  motion.  When  water  flows  in  a pipe,  con- 
duit, or  channel  of  any  kind,  the  velocity  is  not  the  same  at 
all  points  of  the  flow,  unless  all  cross-sections  of  the  pipe  or 
channel  are  equal.  That  velocity  which,  being  multiplied  by 
the  area  of  the  cross-section  of  the  stream,  will  equal  the 
total  quantity  discharged,  is  called  the  mean  velocity. 


146 


MECHANICS. 


Let  Q = quantity  that  passes  any  section  in  1 second; 

A = area  of  the  section; 

v = mean  velocity  in  feet  per  second. 

Then,  Q = Av,  and  v = 

The  vertical  distance  between  the  level  surface  of  the 
water  and  the  center  of  the  aperture  through  which  it  flows, 
is  called  the  head. 

Let  V = mean  velocity  of  efflux  through  a small  aperture; 

h = head  in  feet  at  the  center  of  the  aperture; 

tv  = weight  of  water  flowing  through  the  aperture 
per  second. 

Then,  V = 1/2  g h;  that  is,  the,  velocity  of  efflux  is  the 
same  as  if  the  water  had  fallen  through  a height  equal  to 
the  head. 

Let  Q = theoretical  number  of  cubic  feet  discharged  per 
second; 

Vm  = mean  velocity  through  orifice  in  feet  per  second; 

A = area  of  orifice; 

h = theoretical  head  necessary  to  give  a mean 
velocity  Vm; 

Qa  = actual  quantity/discharged  in  cubic  feet  per 
second. 

Then,  for  an  orifice  in  a thin  plate,  or  a square-edged 
orifice  (the  hole  itself  may  be  of  any  shape,  triangular,  square, 
circular,  etc.,  but  the  edges  must  not  be  rounded),  the  actual 
quantity  discharged  is 

Qa  = .615  Q = .615  A Vm. 

The  weir  is  a device  used  for  measuring  the  discharge  of 
water.  It  is  a retangular  orifice  through  which  the  water 
flows. 

If  d = the  depth  of  the  opening  in  feet,  and  b its  breadth 
in  feet,  the  area  of  the  opening  is  A = d X b,  and  the  theo- 
retical  discharge  is  Q = dxb  X Vm  = db  X \ V % 9 d,  the 
nead  for  this  case  being  taken  as  d. 

The  actual  discharge  when  the  top  of  the  weir  lies  at  the 
surface  of  the  water  is  . 

Qa  = .615  Q =. 615  Xd5Xf  l/27d  = .615  XI  & l/  2gd3  = 
3.2885  1/  d3. 


HYDROMECHANICS. 


147 


If  h\  is  the  depth  in  feet  of  the  top  of  a weir  below  the 
surface  of  the  water,  and  h is  the  depth  in  feet  of  the  bottom 
of  the  weir  below  the  surface  of  the  water,  the  actual  dis- 
charge Qay  in  cubic  feet  per  second,  is 
Qa  = .615X1  b V^9WW-VW)  = 3.288  b 


FLOW  OF  WATER  IN  PIPES. 


Let  Vm  = mean  velocity  of  discharge  in  feet  per  second; 
h — total  head  in  feet  = vertical  distance  between 
the  level  of  water  in  reservoir  and  the  point 
of  discharge; 

l = length  oDpipe  in  feet; 
d = diameter  or  pipe  in  inches; 
f = coefficient  of  friction. 

Then,  for  straight  cylindrical  pipes  of  uniform  diameter, 
the  mean  velocity  of  efflux  may  be  calculated  by  the 
formula, 

Vm  = 2.315  - 


' Vj 


hd 


' fl  +.125  d‘ 


(a) 


Note.— The  head  is  always  taken  as  the  vertical  distance 
between  the  point  of  discharge  and  the  level  of  the  water  at 
the  source,  or  point  from  which  it  is  taken,  and  is  always 
measured  in  feet.  It  matters  not  how  long  the  pipe  is— 
whether  vertical  or  inclined,  whether  straight  or  curved,  nor 
whether  any  part  of  the  pipe  goes  below  the  level  of  the 
point  of  discharge  or  not— the  head  is  always  measured  as 
stated  above. 


Example.— What  is  the  mean  velocity  of  efflux  from  a 
6"  pipe,  5,780  ft.  long,  if  the  head  is  170  ft.  ? Take  / = .021. 
Solution  — 

Vm  = 2.315  yjfl  + md  = 2.315  ^021  x 5,730  + (.125  X 
=^=  6.69  ft.  per  sec. 

When  the  pipe  is  very  long  compared  with  the  diameter, 
as  in  the  above  example,  the  following  formula  may  be 
used:  y 

Vm  = 2.315  "y  jy’  Jb) 

in  which  the  letters  have  the  same  meaning  as  in  the  prece- 
ding formula.  This  formula  may  be  used  -when  the  length 
of  the  pipe  exceeds  10,000  times  its  diameter. 


148 


MECHANICS. 


The  actual  head  necessary  to  produce  a certain  velocity 
Vm  may  be  calculated  by  the  formula 

+ -0233  v*  (0 

If  the  head,  the  length  of  the  pipe,  and  the  diameter  of  the 
pipe  are  given,  to  find  the  discharge,  use  the  formula 

« = ' OS445  <*2V/ITl25d:  W 

that  is,  the  discharge  in  gallons  per  second  equals  .09445 
times  the  square  of  the  diameter  of  the  pipe  in  inches,  multi- 
plied by  the  square  root  of  the  head  in  feet,  multiplied  by  the 
diameter  of  the  pipe  in  inches,  divided  by  the  coefficient  of 
friction  times  the  length  of  the  pipe  in  feet,  plus  .125  times 
the  diameter  of  the  pipe  in  inches. 

To  find  the  value  of  /,  calculate  Vm  by  formula  ( b ) assu- 
ming that  / = .025,  and  get  the  final  value  of  * from  the 
following  table: 


Vm 

/ 

Vm 

f 

Vm 

/ 

.1 

.0686 

.7 

.0349 

2 

.0265 

.2 

.0527 

.8 

.0336 

3 

.0243 

.3 

.0457 

.9 

.0325 

4 

.0230 

.4 

.0415 

1 

.0315 

6 

.0214 

.5 

.0387 

IK 

.0297 

8 

.0205 

.6 

.0365 

i % 

.0284 

12 

.0193 

Example.— The  length  of  a pipe  is  6,270  ft.,  its  diameter  is 
8 in.,  and  the  total  head  at  the  point  of  discharge  is  215  ft. 
How  many  gallons  are  discharged  per  minute  ? 

Solution.—  

Vm  = 3-315 ^7025X^,270  = 7'67  ft'  PCT  SeC"  nearly' 
Using  the  value  of  / = .0205  for  Vm  = 8 (see  table),  Q = 

21 5 V 8 

^05  X 6,270 +(.125  X 8)  = 22-°3  gal'  P6r  ^ = 
22.03  X 60  = 1,321.8  gal.  per  min. 

If  it  is  desired  to  find  the  head  necessary  to  give  a discharge 
of  a certain  number  of  gallons  per  second  through  a pipe 


HYDROMECHANICS. 


149 


whose  length  and  diameter  are  known,  calculate  the  mean 
velocity  of  efflux  by  using  the  formula 


Vm  = 


24.51  Q . 
d? 


(«) 


findthe  value  of  /from  the  table,  corresponding  to  this  value 
of  Vmi  and  substitute  these  values  of  / and  Vm  in  the  formula 
for  the  head. 

Example.— A 4"  pipe,  2,000  ft.  long,  is  to  discharge  24,000 
gal.  of  water  per  hr.;  what  head  is  necessary? 

24,000  _ _ TT  _ 24.51X61 


Solution.— 


60X60 


= 6f  gal.  per  sec.  Vm 


42 


= 10.2  ft.  per  sec. 

From  the  table,  / = .0205  for  Vm  = 8,  and  .0193  for  Vm 
= 12;  assume  that  / = .02  for  Vm  — 10.2. 

Then,  h = -2-X^^?-22  + .0233  X 10.22  = 196.53  ft. 

To  find  the  diameter  of  a pipe  that  -will  give  any  required 
discharge  in  gallons  per  second,  the  total  length  of  the  pipe 
and  the  head  being  known,  find  the  value  of  d by  formula  (/); 
substitute  this  value  in  formula  (e),  and  find  the  value  of  Vm. 
Then  find  from  the  table  the  value  of  f corresponding  to  this  value 
of  Vm.  Substitute  the  values  of  d and  f just  found  in  the  right- 
hand  member  of  formula  (g)  and  solve  for  d;  the  result  will  be  the 
diameter  of  the  pipe , accurate  enough  for  all  practical  purposes. 


d-1.229^f.  00  d = 2.57^p±M^.  (,) 

Example.— A pipe  2,000  ft.  long  is  required  to  discharge 
24,000  gal.  of  water  per  hr.  The  head  being  195  ft.,  what 
should  be  the  diameter  of  the  pipe? 

Solution.  — Q = = 6§  Sal*  Per  sec-  Substitu- 

ting in  formula  (/),  d = 1.229^j2,000^--—  = 4.18  + in. 

Substituting  this  value  in  formula  (e),  Vm  = — = 

9.352  ft.  per.  sec.  From  the  table,  the  value  of/  for  Vm  = 9.352 
is  .0201.  Substituting  this  value  of  / and  the  value  of  d,  found 
above,  in  formula  (g), 

S _ O /(  0201  x 2,000  -fix  4.18)  X (6f  )2 


150 


STRENGTH  OF  MATERIALS. 


STRENGTH  OF  MATERIALS. 


The  ultimate  strengths  of  different  materials  vary  greatly 
from  the  average  values  given  in  the  following  tables.  In 
actual  practice,  the  safest  procedure  would  be  to  make  a test 
of  the  material  for  its  ultimate  strength  and  coefficient  of 
elasticity,  or  else  specify  in  the  contract  that  it  shall  not  fall 
below  certain  prescribed  limits.  In  the  following  formulas, 

A = area  of  cross-section  of  material  in  square  inches; 

E = coefficient  of  elasticity  in  pounds  per  square  inch; 

O2  = square  of  least  radius  of  gyration; 

I — moment  of  inertia  about  an  axis  passing  through 
the  center  of  gravity  of  the  cross-section; 

M = maximum  bending  moment  in  inch-pounds; 

P = total  stress  in  pounds; 

P = moment  of  resistance; 

S = ultimate  stress  in  lb.  per  sq.  in.  of  area  of  section; 

W = weight  placed  on  a beam  in  pounds; 
b = breadth  of  cross-section  of  beam  in  inches; 
d = depth  of  beam  (in.)  = diam.  of  circ.  section  = alti- 
tude of  triangular  section  = length  of  vertical  side; 
e = amount  of  elongation  or  shortening  in  inches; 

/ = factor  of  safety; 

l = length  in  inches; 

p = pressure  in  pounds  per  square  inch; 

7 r ==  ratio  of  circumference  to  diameter  = 3.1416,  nearly; 
q = a constant  used  in  formula  for  columns; 
r = radius  of  a circular  section; 

s = elastic  set  or  deflection  in  inches  of  a beam  under  a 
. transverse  (bending)  stress; 
t — thickness  of  a shell  or  hollow  section. 

For  tension,  compression  (where  the  piece  does  not  exceed 
10  times  its  least  diameter),  and  shear, 


To  find  the  breaking  stress  (P),  make/  = 1.  For  safe  load, 
take  / from  Table  I,  and  S from  Table  II,  according  to  the 
nature  and  character  of  stress. 


STKENGTH  TABLES. 


151 


TABLE  I. 

Factors  of  Safety  (/). 


Name  of  Material. 

Steady 

Stress. 

Varying 

Stress. 

Shocks 

(Ma- 

chines). 

Cast  iron  

6 

15 

20 

Wrought  iron 

4 

6 

10 

Steel 

5 

7 

15 

Wood  

8 

10 

15 

Brick  and  stone 

15 

25 

30 

TABLE  II. 


Ultimate  Strengths  (S). 


Name  of  Material. 

Tension. 

Com- 

pression. 

Shear. 

Flexure. 

Cast  iron 

Wrought  iron 

Steel  

Wood 

Stone 

20,000 

50.000 
100,000 

10.000 

90.000 

50.000 
150,000 

8,000 

6,000 

2,500 

20,000 

47.000 

70.000 
600  to  3,000 

36.000 

50.000 
120,000 

9.000 

2.000 

Brick 

200 

Example— A square  cast-iron  pillar  18  in.  long  is  required 
to  sustain  a steady  load  of  75,000  lb.;  what  must  be  the  length 
of  a side  ? 

Solution.— From  the  table,  / = 6,  and  5 = 90,000.  By 
formula  (1), 


AS 

f ' 


Pf 

S 


75,000  X 6 
90,000 


5 sq. in. 


Length  of  side  = j/ 5 = 2.236  in.,  say  2|  in. 


The  amount  of  elongation  or  of  shortening  of  a piece  under 
a stress  is  given  by  the  formula 


e 


PI 

AE' 


(2) 


The  coefficient  of  elasticity  ( E ) must  be  taken  from  the 
following  table: 


152 


STRENGTH  OF  MATERIALS. 


TABLE  III. 


Name  of  Material. 

Coefficient 
of  Elasticity. 

Elastic  Limit 
for  Tension. 

Cast  iron 

15,000,000 

6,000 

Wrought  iron 

25,000,000 

25,000 

Steel  

30,000,000 

50,000 

Wood 

1,500,000 

3,000 

A wrought-iron  bar  24  ft.  long,  1£  in.  in  diameter,  would 
elongate,  under  a tensile  stress  of  15  tons, 

(15  X 2,000)  X (24  X 12)  _ 
i 77  (H)2  X 25,000,000 

To  find  the  breaking  strength  of  a beam , use  the  formula 
M = SR.  (3) 

Obtain  M and  R from  the  two  following  tables,  according 
to  the  kind  of  beam  and  nature  of  cross-section.  A simple 
beam  is  one  merely  supported  at  its  ends.  In  the  expression 
for  R,  d is  always  understood  to  be  the  vertical  side  or  depth; 
hence,  that  beam  is  the  stronger  which  always  has  its 
greatest  depth  or  longest  side  vertical.  The  moment  of 
inertia  I is  taken  about  an  axis  perpendicular  to  d,  and 
lying  in  the  same  plane. 


TABLE  IV. 


Kind  of  Beam  and  Man- 
ner of  Loading. 

Bending 

Moment. 

M 

Cantilever,  load  at  end  

Wl 

Cantilever, uniformly  loaded 

A Wl 

Simple  beam,  load  at  mid- 
dle   

A Wl 

Simple  beam,  uniformly 
loaded  

Vs  Wl 

Beam  fixed  at  both  ends, 
load  at  middle 

Vs  Wl 

Beam  fixed  at  both  ends, 
uniformly  loaded  

A Wl 

Deflection. 

s 


STRENGTH  OF  MATERIALS. 


153 


TABLE  V. 


Name  of  Section. 


I 


R 


£2 


Solid 

circular 


7i d4 


nd* 

32 


d2 

16 


Hollow 

circular 


7r(d4_^14) 

64 


n(d^-d^) 
32  d 


d2  + dx2 
16 


Solid  square 


Hollow 

square 


d4 

12 

d4  — dx4 
12 


d3 

6 

d4  — dx4 
6d 


d2 

12 

d2  + dx2 
12 


Solid 

rectangular 


Hollow 

rectangular 


5d3 

12 


fcd2 

6 


b2 

12 


ftdS-M!3 

12 


bd3—bidi3 

6d 


63d  - &i3di 
12  (6d  — 6idi) 


Solid 

triangular 


fcd3 

36 


fed2 

24 


d2 

18 


Solid 

elliptical 


nbd3 

64 


_jr6d2 

32 


16 


Hollow 

elliptical 


I-beam 
Cross  with 
equal  arms 
(approxi- 
mate) 
Angle  with 
equal  arms 
(approxi- 
mate) 


^(&d3-Mi3) 


7r(6d3-6idi 

32d 


3 63d  - b^di 

16(bd  — 6idi) 


6d3-61d13 

12 


6d3-6xdi3 

6d 


63d  - &!3d! 
12  (6d  — Mi) 


d2 

22.5 


d2 

25 


154 


STRENGTH  OF  MATERIALS. 


Thus,  the  breaking  strength  of  a cast-iron  simple  beam 
uniformly  loaded  and  20  ft.  long  between  the  supports,  hav- 
ing a hollow  rectangular  cross-section  8 in.  by  6 in.  outside 
and  6 in.  by  4 in.  inside,  is  given  by  the  formula 


M = SR, or  \Wl  = 36,000  X — .&1  ^ 
o CL 

Using  a factor  of  safety  of  6,  the  beam  should  support 


55,200 


- = 9,200  lb. 


with  perfect  safety.  The  value  of  S for  beams  should  be 
taken  from  the  flexure  column  of  Table  II. 

To  find  the  amount  of  deflection  in  a beam  due  to  a load,  sub- 
stitute the  values  of  Wf  l,  E,  and  I in  the  different  expres- 
sions for  the  deflection  s in  Table  IV. 

The  value  of  I is  to  be  taken  from  Table  V. 

Example.— What  is  the  deflection  of  a wrought-iron  beam 
fixed  at  both  ends,  7 ft.  long  between  the  supports,  having  a 
solid  rectangular  cross-section  6 in.  wide  and  2£  in.  deep, 
carrying  a load  of  21,000  lb.  in  the  middle  ? 

Solution.— From  the  table, 

WP  WP  21,000  X (7  X 12)3  X 12  _ 

S 192  El  6d3  192 X 25,000,000 X 6 X (2£)3  * 

192  Ji.  X 12 

Example.— It  is  desired  to  calculate  the  depth  (d)  of  a 
cast-iron  cantilever  36  in.  in  length  (=  l)  that  will  sustain 
at  its  end  a weight  of  4,000  lb.  (=  W),  the  lever  to  be  of  rect- 
angular section  and  2 in.  in  width. 

Solution.— The  ultimate  stress  per  square  inch  for  cast 
iron  in  flexure  is  given  in  Table  II  as  36,000  lb.  (=5). 
The  weight  will  be  a steady  load,  and  therefore,  according 
to  Table  I,  a factor  of  safety  of  6 should  be  used.  By  for- 
mula (3),  M = SR.  For  a cantilever  beam  carrying  a load 
at  the  end,  M = Wl  (Table  IV);  and  for  a rectangular  sec- 


tion, R = (Table  V). 

D 

Then,  as  W = 4,000,  l = 36,  b = 2,  / = 6,  we  have 


STRENGTH  OF  BEAMS. 


155 


The  value  of  d is  found  by  substituting  in  this  equation 
the  known  values  of  S,  b , W,  l,  and/,  as  follows: 

v d = 4,000  X 36;  whence,  d = 8.49  in. 
o X o 

At  the  point  where  the  beam  is  supported,  the  required 
depth  is  found  to  be  8.49,  or,  practically,  8£  in.  At  a point 
6 in.  from  the  support,  the  depth  may  again  be  calculated  by 
substituting  in  the  equation  the  value  of  l (the  overhanging 
length  beyond  this  point);  l = 30,  and  the  equation  becomes 

a xg.  4,000  X 80. 

b X o 

d = 7.75  in. 

At  a point  12  in.  from  the  support,  l = 24,  and 
36,000  X 2 X d? 

6X6 


= 4,000  X 24;  whence,  d = 6.93  in. 


At  a point  18  in.  from  the  support,  l = 18;  and  from  the 
equation,  d = 6 in.;  at  24  in.  from  the  support,  l = 12  and 
d = 4.9  in.;  at  30  in.  from  the  support,  1=6  and  d = 3.46 in.; 
at  36  in.  from  the  support,  or  at  the  end  of  the  beam,  l = 0 
and  d = 0. 


The  depths  required  to  be  given  to  the  lever  or  beam  at 
the  point  of  support  and  at  intervals  of  6 inches  along  its 


length,  are  found  to  be  8.49,  7.75,  6.93,  6,  4.90,  and  3.46 
inches,  respectively. 

The  lever  is  shown  in  the  figure;  theoretically,  it  would 
taper  to  nothing  at  the  end,  as  indicated  by  dotted  lines,  but 
practically  sufficient  metal  must  be  added  at  that  point  to 
provide  means  of  attaching  the  weight. 


156 


STRENGTH  OF  MATERIALS. 


Note.— In  the  preceding  examples  the  weight  of  the  beam 
has  been  neglected.  If,  however,  this  weight  is  large  in  com- 
parison with  the  weight  or  weights  carried  by  the  beam,  it 
should  be  taken  into  account,  considering  it  (when  the  cross- 
section  of  the  beam  is  the  same  throughout)  as  a load  uni- 
formly distributed  over  the  whole  length  of  the  beam. 


COLUMNS. 

To  find  the  breaking  strength  of  a column,  use  the  follow- 
ing formula: 


P = 


SA 

V * 
1 + q GP 


(4) 


S is  taken  from  Table  II,  in  the  column  for  compression,  G 2 
from  Table  V,  and  q from  the  following  table,  according  to  the 
character  of  the  ends. 


TABLE  VI. 


Material. 

Both  Ends 

One  End 

Both  Ends 

Flat  or  Fixed. 

Round. 

Round. 

Cast  iron 

1 

1.78 

4 

5,000 

5,000 

5,000 

Wrought  iron 

1 

1.78 

4 

36,000 

36,000 

36^000 

Steel 

1 

1.78 

4 

25,000 

25,000 

25,000 

Wood 

1 

1.78 

4 

3,000 

3,000 

3,000 

The  breaking  load  of  an  elliptical  wooden  column  18  ft. 
long,  having  rounded  ends,  the  diameters  of  the  cross-section 
being  12  in.  and  8 in.,  is 

SA  _ 8,000  X (i  it  X 12  X 8) 


P = 


4 (18  X 12) 2 

3,000  82 


= 36,442  lb. 


16 


Using  a factor  of  safety  of  8,  the  column  should  support 
36,442 
8 


! = 4,565  lb.  with  perfect  safety. 


SHAFTING. 


157 


SHAFTING. 

The  diameter  of  a shaft  may  be  found  by  the  following 
formulas.  The  first  is  used  when  great  stiffness  is  required, 
and  the  shafts  are  very  long;  the  second  when  strength  only 
is  required  to  be  considered. 


d = diameter  of  shaft  in  inches; 

H = horsepower  transmitted; 

N = number  of  revolutions  per  minute; 
c = constant  in  formula  (5); 
k = constant  in  formula  (6). 


c = 5.26  for  cast  iron;  4.75  for  wrought  iron;  3.96  for  steel; 
k = 4.02  for  cast  iron;  3.63  for  wrought  iron;  3.03  for  steel. 
Note.— To  extract  the  fourth  root,  extract  the  square 
root  twice. 


p = pressure  in  pounds  per  square  inch; 
d = diameter  of  pipe  or  cylinder  in  inches; 
t = thickness  in  inches; 

S = ultimate  tensile  strength  taken  from  Table  II; 
r = inside  radius  in  inches; 

/ = factor  of  safety,  usually  taken  as  6 for  wrought  iron 
and  12  for  cast  iron. 

For  thin  pipes,  pdf  = 2 tJS.  (7) 

For  thick  pipes  or  cylinders, 


D = diameter  of  the  rope  in  inches  = diameter  of  iron 
from  which  the  link  in  chain  is  made; 

W = safe  load  in  tons  of  2,000  lb. 

For  common  hemp  rope,  W = £ D2. 

For  iron-wire  rope,  W = § D2. 

For  steel-wire  rope,  W = D2. 

For  close-link  wrought-iron  chain,  W — 6 D2. 

For  stud-link  wrought-iron  chain,  W = 9 D2. 


(5)  d = k^j§.  (6) 


PIPES  AND  CYLINDERS. 


ROPES  AND  CHAINS. 


158 


BOILERS. 


BOILERS. 


BOILER  DESIGN. 


TO  DEVELOP  THE  DOME  OF  A BOILER. 

A side  view  of  the  dome,  together  with  a section  of  the 
boiler,  is  shown  in  Fig.  A.  Draw  Fig.  B,  the  end  view  of 
the  dome  and  of  the  boiler.  Above  the  dome  draw  a circle 
ine"  m of  the  same  diameter  as  the  dome.  Divide  the  lower 


half  of  this  circle,  as  n e"  m,  into  any  number  of  equal  parts, 
as  me",  c"  d",  d"  e",  e"f",  and/"  g".  The  greater  the  num- 
ber of  these  divisions,  the  more  accurate  will  be  the  results. 
From  the  points  of  division  c",  d ",  e",/",  and  g",  draw  lines 
parallel  to  the  vertical  center  line  of  the  boiler,  as  c"  c',  d"  d\ 
/"/',  and  g"  g'. 

We  are  now  ready  to  draw  the  templet  of  the  dome,  as 
shown  in  Fig.  C.  Draw  a straight  line  of  indefinite  length, 
and  on  it  lay  off  a distance  h i equal  to  the  circumference  of 


BOILER  DESIGN. 


159 


the  dome.  (The  circumference  of  the  dome  is  found  by 
multiplying  the  diameter  a b of  the  dome  by  3.1416.)  Divide 
the  distance  h i into  twice  the  number  of  equal  parts  that  the 
semicircle  above  the  dome  in  Fig.  B has.  In  the  figure  it  has 
been  divided  into  6 equal  parts;  therefore,  divide  this  line 
into  2 X 6 = 12  equal  parts,  as  bg,  gf,  fe , ed , etc.,  and 
through  these  points  of  division  draw  lines  at  right  angles  to 
the  line  hi,  as  shown;  make  the  length  of  each  of  these  lines 
the  same  as  the  length  of  the  line  that  corresponds  to  it  in 
Fig.  B.  Thus,  e e'  is  equal  to  e e'  in  Fig.  B,  d d ' is  equal  to 
d d'  in  Fig.  B,  a a'  is  equal  to  a a'  in  Fig.  B,  etc.  After  hav- 
ing laid  off  the  lengths  of  these  lines,  draw  the  curved  line 
i'  c'  h'.  This  being  done,  we  have  the  templet  of  the  dome 
on  the  seam.  The  lap  for  riveting  must  be  allowed,  as  shown 
by  the  dotted  lines  around  the  templet. 


TO  DEVELOP  THE  SLOPE  SHEET  abed  OF  A BOILER, 
SHOWN  AT  A IN  THE  FIGURE  BELOW. 


Draw  a straight  line  a b,  as  shown  in  Fig.  B,  and  on  it  lay 
off  the  distance  ad,  equal  to  be,  Fig.  A.  At  a and  d,  erect 


perpendiculars  a c and  d e , respectively,  making  a c equal  to 
b a,  and  d e equal  to  c d,  of  Fig.  A.  With  a point  6 on  a 6 as  a 
center,  and  a radius  d e,  describe  the  quadrant/ g.  Divide  this 
quadrant  into  any  number  of  parts;  the  greater  the  number, 


160 


BOILERS. 


the  more  accurate  will  he  the  results.  Here  it  is  divided 
into  three,  as  g-i,  1-2,  and  2-f.  Through  the  points  g,  i,  and  2, 
draw  lines  parallel  to  a o,  intersecting  the  perpendicular  d e 
in  e,  1',  and  2',  and  the  perpendicular  bginh  and  i.  Througn 
the  points  V,  2 ',  and  d,  draw  lines  parallel  to  c e.  Through  any 
point,  as  J , on  the  line  ce,  draw  JK  perpendicular  to  ce,  cut- 
ting the  lines  2"-2',  and  3"-d  in  the  points  i,  n , and  K , 
respectively.  From  the  line  J K lay  off  the  distances  i m,  n o, 
and  Kp,  equal  to  the  distances  hi,  i 2,  and  bf,  respectively, 
and  pass  the  dotted  curve  Jmop  through  the  points.  Now 
draw  Fig.  C.  Draw  the  straight  line  kq,  and  through  the 
point  J draw  ec  perpendicular  to  it.  Lay  off  on  the  line  kq, 
on  each  side  of  the  line  c e,  points  ra'  and  m'  at  distances 
from  it  equal  to  the  length  of  Jm  in  Fig.  B.  Lay  off,  also, 
points  o'  and  o'  at  distances  from  m'  and  m'  equal  to  m o in 
Fig.  B\  also,  points p'  and  p'  at  distances  from  o'  and  o'  equal 
to  op  of  Fig.  B.  Through  the  points  thus  laid  off,  draw  lines 
parallel  to  c e.  Lay  off  the  distances  J c and  J e from  J,  in 
Fig.  C,  equal  to  Jc  and  Je , respectively,  in  Fig.  B\  the  dis- 
tances m'  V"  and  m'  1"  from  m'  equal  to  i 1 " and  i V in 
Fig.  B;  o'  2"'  and  o'  2"  from  o'  equal  to  n2"  and  n2 ';  and 
pf  3'"  and  p'  3"  from  p'  equal  to  K3"  and  Kd  of  Fig.  B. 
Through  the  points  thus  laid  off  draw  the  curved  lines 
S'"  c3'"  and  3"  e 3".  With  the  points  3"  as  centers  and  a 
radius  a d,  Fig,  B,  describe  the  arcs  r and  r.  With  the  points 
S'"  as  centers  and  a radius  3"  a,  Fig.  B,  describe  the  arcs 
s and  s.  From  the  points  of  intersection  of  these  arcs,  draw 
lines  to  the  points  3'"  and  3".  This  being  done,  we  have  the 
templet  of  the  slope  sheet  on  the  seams.  The  laps  for  rivet- 
ing must  be  allowed  as  shown  by  the  dotted  lines  around  the 
templet.  _____ 

TO  DEVELOP  THE  SLOPE  SHEET  ImnO  OF  A BOILER, 
SHOWN  AT  A IN  THE  FIGURE  ON  THE 
FOLLOWING  PAGE. 

Draw  the  two  views  of  the  sheet  as  shown  in  Figs.  B and  C. 
Suppose  the  seam  to  be  at  o n,  Fig.  A,  and  the  sheet  to  be 
made  in  one  piece.  Divide  the  semicircles  a dg  and  a'  d'g', 
Fig.  C,  into  any  number  of  equal  parts;  the  greater  the  number 


BOILER  DESIGN. 


161 


of  these  divisions,  the  more  accurate  will  be  the  results. 
Join  the  points  b and  b',  c and  c',  d and  df,  e and  e',  and  / and 
f by  full  lines,  and  join  the  points  b and  a',  c and  b ',  d and  c', 
e and  d',  / and  e',  and  g and  f by  dotted  lines,  as  shown. 
Then  draw  Figs.  D and  E.  Draw  at  right  angles  to  one 
another  the  lines  wa  and  wx , also  the  lines  za'  and  zy. 
Make  the  length  of  the  line  wx  equal  to  r,  Fig.  B,  and  the 


length  of  the  line  w a equal  to  a a',  Fig.  C.  From  w lay  off  on 
the  line  wa,  Fig.  D,  distances  wb,  w c,  w d,  w e,  w f,  and  wg, 
respectively,  equal  to  the  lengths  of  the  full  lines  bb',  cc\ 
etc.  of  Fig.  C,  and  draw  the  lines  ax,  bx,  cx,  dx,  ex,  fx,  and 
gx,  as  shown.  Make  the  length  of  the  line  zy,  Fig.  E,  the 
same  as  that  of  w x,  Fig.  D.  From  z lay  off  on  the  line  z a\ 


162 


BOILERS. 


Fig.  E,  distances  za zb zc\  zd',  ze ',  and  zf,  respectively, 
equal  to  the  lengths  of  the  dotted  lines  ba',  cb',  etc.,  in 
Fig.  C,  and  draw  the  lines  a'  y , b ’ y,  c'  y,  f y,  d ' y,  and  e'  y. 

We  are  now  ready  to  draw  the  templet  of  the  slope  sheet. 
Instead  of  drawing  the  whole  templet,  we  will  draw  only 
one-half  of  it,  as  is  shown  in  Fig.  F,  since  the  other  half  is 
exactly  the  same.  Draw  the  line  a a',  and  make  it  equal  in 
length  to  the  distance  ax,  Fig.  D.  With  a'  as  a center,  and  a 
radius  ya',  Fig.  E,  describe  an  arc  at  b.  With  a as  a center 
and  a radius  = arc  a b,  Fig.  C,  describe  another  arc  inter- 
secting the  first  arc  in  b.  With  a'  as  a center,  and  a radius 
= arc  a'  b',  Fig.  C,  describe  an  arc  at  b'.  With  b as  a center, 
and  a radius  x b,  Fig.  D,  describe  an  arc,  intersecting  the  arc 
already  drawn,  at  b';  draw  the  full  line  bb ' and  dotted  line 
b a'.  With  b'  as  a center,  and  a radius  y b',  Fig.  E,  describe 
an  arc  at  c.  With  & as  a center,  and  a radius  = arc  c b,  Fig.  C, 
describe  an  arc  cutting  the  last  arc  at  c.  With  b ' as  a center, 
and  a raditis  = arc  c'b ',  Fig.  C,  describe  an  arc  at  c'.  With  c 
as  a center,  and  a radius  x c,  Fig.  D,  describe  an  arc  cutting 
the  last  arc  at  c';  draw  the  full  line  cc’  and  dotted  line  cb'. 

Continue  to  construct  the  remaining  portion  of  the  half 
templet  in  a similar  manner,  taking  the  distances  for  the  full 
lines  from  Fig.  D,  and  those  for  the  dotted  lines  from  Fig.  E. 
Through  the  points  a,  b,  c,  d , e,  /,  and  g,  and  through  the 
points  a',  b',  c',  d',  e',  f,  and  g',  draw  the  curved  lines  shown. 
Since  this  is  the  development  of  the  slope  sheet  at  the  seam, 
the  laps  for  riveting  must  be  allowed;  they  are  shown  by  the 
dotted  lines  around  the  templet  in  Fig.  F. 


CARE  AND  INSPECTION  OF  BOILERS. 


POINTS  TO  BE  OBSERVED. 

Preliminary  to  a boiler  inspection,  the  boiler,  flues,  mud- 
drum,  ash-pit,  and  all  connections  should  be  thoroughly 
cleaned,  to  facilitate  a careful  examination.  Blisters  may 
occur  in  the  best  iron  or  steel,  and  their  presence,  and  also 
that  of  thin  places,  is  ascertained  by  going  over  all  parts  ot 
the  boiler  with  a hammer.  When  blisters  are  discovered,  the 
plates  should  be  repaired  or  replaced.  Repairing  a blister 


CARE  OF  BOILERS. 


163 


consists  in  cutting  out  the  blistered  space  and  riveting  a 
“hard  patch”  over  the  hole  on  the  inside  of  the  boiler,  if 
possible,  to  avoid  forming  a pocket  for  sediment.  All  seams, 
heads,  and  tube  ends  should  be  examined  for  leaks,  cracks, 
corrosions,  pitting,  and  grooving,  detection  of  the  latter 
possibly  requiring  the  use  of  a magnifying  glass.  Uniform 
corrosion  is  a wasting  away  of  the  plates,  and  its  depth  can  be 
determined  only  by  drilling  through  the  plate  and  measuring 
the  thickness,  afterwards  plugging  the  hole.  Pitting  is  due 
to  a local  chemical  action,  and  is  readily  perceived.  Grooving 
is  usually  due  to  buckling  of  the  plates  when  under  pressure, 
and  frequently  to  the  careless  use  of  the  sharp  calking  tool. 
Seam  leaks  are  generally  caused  by  overheating,  and  demand 
careful  examination,  as  there  may  be  cracks  under  the  rivet 
heads.  If  such  cracks  are  discovered,  the  seam  should  be  cut 
out,  and  a patch  riveted  on.  Loose  rivets  should  be  carefully 
looked  for,  and  should  be  cut  out  and  replaced,  if  found. 
Pockets,  or  bulging,  and  burns  should  be  looked  for  in  the 
firebox.  The  former  are  not  necessarily  dangerous,  but  if 
there  are  indications  of  their  increasing,  they  should  be  heated 
and  forced  back  into  place  or  cut  out  and  a patch  put  on. 
Burns  are  due  to  low  water,  the  presence  of  scales,  or  to  the 
continuous  action  of  flames  formed  on  account  of  air  leaking 
through  the  brickwork.  The  burned  spots  should  be  cut  out 
and  patched  as  previously  described.  The  conditions  of  all 
stays,  braces,  and  their  fastenings  should  be  examined  and 
defective  ones  replaced.  The  shell  of  the  boiler  should  be 
thoroughly  examined  externally  for  evidences  of  corrosion, 
which  is  liable  to  set  in  on  account  of  dampness,  exposure  to 
weather,  leakage,  etc.,  and  may  be  serious.  The  boiler  should 
be  so  set  that  joints  and  seams  are  accessible  for  inspection, 
and  should  have  as  little  brickwork  in  contact  with  it  as 
possible.  The  brickwork  should  be  in  good  condition,  and 
not  have  air  holes  in  it,  since  they  decrease  the  efficiency  of 
the  boiler  and  are  liable  to  cause  injury  to  the  plates  by 
burning,  as  above  explained,  and  also  by  unevenly  heating 
and  distorting  them.  The  mud-drum  and  its  connections 
are  liable  to  corrosion,  pitting,  and  grooving,  and  should  be 
examined  as  carefully  as  the  boiler. 


164 


BOILERS. 


All  valves  about  a boiler  should  be  easy  of  access,  and 
should  be  kept  clean  and  working  freely.  Each  boiler  should 
have  at  least  three  gauge-cocks,  properly  located,  and  it  is  of 
the  utmost  importance  that  they  be  kept  clean  and  in  order, 
and  the  same  may  be  said  of  the  glass  water  gauge.  The 
middle  gauge-cock  should  be  at  the  water  level  of  the  boiler, 
and  the  other  two  should  be  placed  one  above  and  one  below 
it,  at  a distance  of  about  6 in. 

The  condition  of  the  pumps  or  injectors  should  be  looked 
into  to  make  sure  that  they  are  in  the  best  working  order.  The 
steam  gauge  should  be  tested  to  ascertain  that  it  indicates 
correctly,  and  if  it  does  not,  it  should  be  corrected.  If  the 
hydraulic  test  is  to  be  used,  the  boiler  should  be  tested  to  a 
pressure  of  50fo  higher  than  that  at  which  the  safety  valve  will 
be  set. 

External  Inspection  When  Boiler  Is  Under  Steam.— The  gauge- 

cocks,  and  also  the  gauge  glass,  should  be  tried,  to  make  sure 
that  they  are  not  choked.  The  steam  gauge  should  be  taken 
down,  if  permissible,  and  tested,  and  corrected  if  necessary. 
The  gauge  pointer  should  move  freely.  Blowing  out  the 
gauge  connection  will  show  whether  it  is  clear  or  not.  The 
boiler  connections  should  be  examined  for  leaks.  The  safety 
valve  should  be  lifted  from  its  seat,  to  make  sure  that  it  does 
not  stick  from  any  cause,  and  it  should  be  seen  that  the 
weight  is  in  the  right  place.  Observe  from  the  steam  gauge 
if  the  valve  blows  off  at  the  pressure  it  is  set  for.  See  that  all 
pumps  and  feed-apparatus  are  working  properly,  and  that  the 
blow-off  and  check-valves  are  in  order.  Blisters  and  bagging 
may  sometimes  be  detected  in  the  furnace.  The  condition  of 
the  brickwork  is  of  considerable  importance,  since  the 
existence  of  air  holes  is  a source  of  trouble,  as'  already 
explained. 

Incrustation. — One  of  the  chief  sources  of  trouble  to  the 
boiler  user  is  that  of  incrustation.  All  wTater  is  more  or  less 
impure;  and  as  the  water  in  the  boiler  is  continuously  evapo- 
rated, the  impurities  are  left  behind  as  powder  or  sediment. 
This  collects  on  the  plates,  forming  a scaly  deposit,  varying 
in  nature  from  a spongy,  friable  texture  to  a hard,  stony  one. 
This  deposit  impedes  the  transmission  of  heat  from  the  plates 


CARE  OF  BOILERS. 


165 


to  the  water  and  often  causes  overheating  and  injury  to  the 
plates.  It  is  probable  that  ^ in.  of  scale  necessitates  the  con- 
sumption of  12$  to  20$  more  fuel.  The  various  impurities  in 
the  water  may  be  either  in  suspension  or  solution.  If  the 
former,  the  water  can  be  purified  by  filtration  before  going 
into  the  boiler.  If  the  latter,  the  substances  must  first  be 
precipitated  and  then  filtered.  Many  impurities  (sulphate 
and  carbonate  of  lime,  etc.)  may  be  removed  by  heating  the 
water  before  feeding  it  into  the  boiler. 

The  first  thing  to  do,  when  dealing  with  a water  supply,  is 
to  have  an  analysis  of  it  made  by  a competent  chemist.  The 
fact  that  a water  contains  a certain  amount  of  solid  matter  is 
no  criterion  as  to  its  unfitness  for  boiler  use.  The  presence  of 
certain  salts,  as  carbonate  or  chloride  of  sodium,  even  in  large 
quantities  (say  40  to  50  gr.  per  gal.),  would  not  be  serious  if 
due  attention  were  given  to  the  blowing  off.  On  the  other 
hand,  salts  of  lime  in  the  above  proportion  would  be  very 
objectionable,  requiring  greatly  increased  attention  in  the 
matter  of  purification  and  blowing  off  or  else  cleaning  out. 

The  various  methods  of  dealing  with  impure  water  may  be 
classed  as  follows: 

1.  Filtration.— Where  the  matter  (sand,  mud,  etc.)  is  held 
in  suspension,  it  can  be  removed,  before  the  water  enters 
the  boiler,  by  the  aid  of  settling  tanks  or  by  filtering,  or  by 
forcing  the  water  up  through  layers  of  sand,  broken  brick, 
etc.,  or  by  using  filtering  cloths  in  a proper  machine. 

2.  Chemical  Treatment. — Clark’s  process,  combined  with  a 
subsequent  filtration  (the  joint  process  being  known  as  the 
Atkins  system),  has  been  successfully  applied  on  both  small 
and  large  scales  in  the  chalk  districts  of  England.  Lime 
water  is  mixed  with  the  water  to  be  purified,  the  amount 
used  depending  on  the  composition  of  the  water,  as  deter- 
mined by  a careful  analysis.  The  lime  is  thus  precipitated, 
and  the  water  is  then  filtered  in  a machine  containing  travel- 
ing cotton  cloths.  Not  only  is  the  carbonate  of  lime  entirely 
removed,  but  it  has  been  proved  that  any  sulphate  of  lime 
that  may  be  present  is  also  prevented  from  incrusting.  This 
is  important,  as  the  latter  impurity  forms,  perhaps,  the  "worst 
scale  one  has  to  contend  with. 


166 


BOILERS. 


Various  chemical  compounds  are  in  use  for  boilers.  Car- 
bonate of  soda  is  perhaps  the  best  general  remedy.  It  forms 
the  basis,  in  fact,  of  nearly  all  boiler  compounds,  whatever 
their  name  or  appearance.  This  soda  deals  efficaciously  both 
with  the  carbonate  and  the  sulphate  of  lime.  The  precipi- 
tates thus  thrown  down  do  not  form  a hard  crust;  they  can 
be  washed  out  in  the  form  of  sludge  or  mud. 

Carbonate  of  soda  is  also  useful  where  condensers  are 
employed,  as  it  counteracts  the  effect  of  the  grease,  which  is 
brought  over  with  the  exhaust  steam.  If  used  in  too  large 
quantities,  it  will  cause  priming.  The  best  way  to  use  it  is  to 
make  a solution  of  it  and  connect  with  the  feed,  fixing  a cock 
so  as  to  regulate  the  amount  fed  in.  Soda  ash  is  cheaper, 
but  more  of  it  is  required,  and,  besides,  it  is  generally  impure. 
Caustic  soda  removes  lime  scale  quicker  than  ordinary  soda 
does,  but  it  is  much  stronger  and  liable  to  attack  the  plates.  It 
should  be  used  in  smaller  quantities  than  the  ordinary  kind. 

Barks,  molasses,  vinegar,  etc.  develop  acids  that  attack  the 
plates.  Animal  and  vegetable  oils  do  the  same,  and  also 
harden  the  deposits  and  make  their  removal  more  difficult. 
It  is  a good  rule  to  keep  all  animal  and  vegetable  matter  out 
of  boilers  altogether. 

Feed-Water  Heaters.— Carbonates  and  sulphates  of  lime  are 
precipitated  by  high  temperatures.  The  heaters  should  be 
arranged  so  that  the  deposit  forms  chiefly  on  a series  of 
plates  that  can  be  easily  removed  for  cleaning.  If -the 
deposit  gathers  in  pipes,  however,  it  is  simply  transferring 
the  evil  from  one  vessel  to  another.  A double  advantage 
is  gained  by  these  heaters,  for  the  feedwater  is  put  into  the 
boiler  already  heated,  and  so  fuel  is  saved. 

Mechanical  Aids.— Deposits  take  place  chiefly  in  sluggish 
places.  Various  devices  to  aid  circulation  have  been  brought 
out.  With  good  attention  and  a not  too  impure  water,  they 
give  satisfactory  results. 

Potatoes,  linseed  oil,  molasses,  etc.  are  sometimes  put  into 
the  boiler  with  the  idea  of  lessening  scale  formation,  by  form- 
ing a kind  of  coating  round  the  particles  of  solid  matter  and 
so  preventing  their  adhering  together.  This  certainly  takes 
place,  but  the  substances  are  injurious,  as  already  pointed  out. 


CARE  OF  BOILERS. 


167 


Whenever  a boiler  has  been  cleaned  out,  we  may  with  advan- 
tage give  the  inside  a thin  coating  of  oil,  or  tallow  and  black 
lead;  this  arrests  the  incrustation  to  a great  extent. 

Sand,  sawdust,  etc.  are  often  used,  the  idea  being  that 
their  grains  act  as  centers  for  the  gathering  together  of  the 
solid  matter  in  the  water,  the  resulting  small  masses  not 
readily  collecting  together  themselves  and  therefore  being 
easily  washed  out.  This  may  be  so,  but  the  cocks,  valves, 
etc.  are  liable  to  suffer  from  the  practice. 

Kerosene  is  strongly  recommended  by  some  boiler  users. 
There  is  no  doubt  that  in  many  cases  its  use  has  given  good 
results.  It  prevents  incrustation,  by  coating  the  particles  of 
matter  with  a thin  covering  of  oil,  the  deposit  thus  formed 
being  easily  blown  out.  The  oil  also  seems  to  act  on  the 
scale  already  formed,  breaking  it  up  and  thus  facilitating  its 
removal.  As  already  remarked,  it  is  a good  plan,  when  the 
boiler  is  empty,  to  give  the  inside  a good  coating  of  this  oil, 
afterwards  putting  it  in  with  the  feed,  the  supply  being  regu- 
lated automatically.  As  to  the  quantity  required,  this  will 
be  found  to  vary  in  different  cases,  according  to  the  nature 
of  the  water;  an  average  of  1 qt.  per  day  for  every  100  horse- 
power will  give  good  results  in  most  cases. 

In  marine  boilers,  strips  of  zinc  are  often  suspended;  the 
deposit  largely  settles  on  them  instead  of  on  the  boiler  plates. 
Also,  any  scale  that  may  be  formed  on  the  latter  is  less  hard 
and  compact  and  more  easily  broken  up.  Further,  any  acids 
formed  by  the  oil  and  grease  brought  over  from  the  con- 
denser attack  this  zinc  instead  of  the  boiler  plates. 

Miscellaneous Acids  are  often  introduced  into  boilers  to 
dissolve  the  scale  already  formed,  the  solid  matter  then  being 
washed  out.  This  treatment  should  be  adopted  with  great 
care,  if  at  all,  as  the  plates  are  likely  to  be  affected. 

Scale  is  often  loosened  and  broken  up  by  deliberately 
inducing  sudden  expansion  or  contraction  in  the  boiler.  In 
the  former  case,  the  expansion  is  brought  about  by  blowing 
off  the  boiler,  and  then,  when  it  is  quite  cooled  down,  turn- 
ing on  steam  at  as  high  a temperature  as  obtainable,  thus 
causing  the  scale  to  expand  more  quickly  than  the  plates  and 
thus  become  loose. 


168 


BOILERS. 


In  the  second  method,  the  boiler  is  blown  off  when  the 
steam  (and  therefore  the  temperature)  is  at  its  highest  and  a 
stream  of  cold  water  then  turned  in.  The  fires  are  then 
drawn  and  the  fire-hole  doors,  dampers,  etc.  opened,  letting 
in  a rush  of  cold  air.  All  this  cools  the  plates  and,  by  the 
contraction  thus  brought  about,  loosens  the  scale.  These 
two  practices  should  be  guarded  against. 

Foaming  or  priming  is  usually  due  either  to  forcing  a 
boiler  beyond  its  capacity  for  furnishing  dry  steam,  or  to  the 
presence  of  foreign  matter.  It  is  dangerous  if  occurring  to 
any  great  extent,  since  water  may  be  carried  along  with 
steam  into  the  engine,  and  a cylinder  head  knocked  out. 
Foaming,  when  it  cannot  be  checked  by  the  use  of  the  sur- 
face blow-out  apparatus,  may  necessitate  the  emptying  of  the 
boiler,  which  must  then  be  filled  with  fresh  water;  this  rids 
the  boiler  of  the  impurities  thdt  have  collected  during  the 
operation  of  the  boiler. 

HORSEPOWER  OF  BOILERS. 

In  actual  practice,  the  result  of  a great  many  tests  has 
shown  that  an  evaporation  of  30  lb.  of  water  per  hr.  from  a 
feedwater  temperature  of  100°  F.  into  steam  at  70  lb.  gauge 
pressure  is  the  equivalent  of  1 horsepower,  or  that  this  steam, 
in  a properly  designed  engine,  will  do  the  equivalent  of 
33,000  X 60  = 1,980,000  ft.-lb.  of  work  per  hr.  In  order,  how- 
ever, to  have  a more  ready  standard  of  comparison,  the  above 
evaporation  has  been  reduced  to  another  standard,  and  is 
found  to  be  equal  to  the  evaporation  of  34.5  lb.  of  water  from 
and  at  a temperature  of  212°  F.  under  atmospheric  pressure, 
and  it  is  on  this  latter  quantity  that  the  calculations  of  the 
horsepower  of  boilers  are  usually  based. 

In  making  an  approximation  of  the  horsepower  of  a given 
boiler,  the  square  feet  of  water-heating  surface  of  the  boiler 
should  first  be  determined,  and  in  doing  this  the  area  of  all 
the  surfaces  exposed  to  the  fire  and  hot  gases,  which,  on 
their  opposite  sides  come  in  contact  with  the  water  in  the 
boiler,  should  be  taken  into  account. 

Example.— An  externally-fired  flue  boiler,  having  a shell 
38  in.  in  diameter,  and  containing  two  flue  pipes  10  in.  in 


HORSEPOWER  OF  BOILERS.  169 

/ 

diameter,  is  22  ft.  long  without  the  smokebox.  If  the  greatest 
depth  of  the  water  in  the  boiler  is  f X 38  = 25.33  in.,  what  is 
the  total  water-heating  area  of  the  boiler  ? 

Solution. — Six  feet  of  the  circumference  of  the  boiler 
shell  lies  below  the  water-line,  as  could  be  found  by  actual 
measurement,  and  the  circumference  of  the  two  flues  is 
equal  to  (1^1416)  x2  = 5.24ft. 


Therefore,  the  water-heating  surface  of  the  shell  is  6 X 22 
= 132  sq.  ft.,  and  that  of  the  flues  is  5.24  X 22  = 115.28  sq.  ft. 
The  w’ater-heating  surface  of  the  heads  of  the  shell  (that  is, 
the  area  below  the  water-line,  minus  the  area  of  the  flues, 
which  could  be  obtained  by  direct  measurement)  is  4.5  X 2 = 
9 sq.  ft.  Therefore,  the  total  water-heating  surface  of  the 
boiler  is  the  sum  of  all  these,  or  256.28  sq.  ft. 

Having  determined  the  water-heating  surface  of  a boiler, 
to  approximate  its  horsepower: 

Rule. — Divide  the  total  water-heating  surface  in  square  feet  by 
the  number  of  square  feet  of  heating  area,  as  given  in  the  table 
below,  required  to  produce  an  evaporation  equivalent  to  1 
horsepower  in  boilei'S  of  the  given  type. 

Example.— The  total  water-heating  surface  of  the  above 
externally-fired  flue  boiler  is  256.28  sq.  ft.  What  is  the  horse- 
power of  the  boiler? 

Solution.— By  referring  to  the  table,  we  find  that  it  takes 
about  10  sq.  ft.  of  heating  surface  to  produce  1 horsepower; 
therefore,  the  above  boiler  would  be  rated  at  about 


= 25.63  H.  P. 


Water-Heating 

Ratio  of  Water- 

Type  of  Boiler. 

Surface  for 
1 Horsepower. 

Heating  Area 
to  Grate  Area 

Square  Feet. 

Required. 

Cylindrical 

Flue 

9 

From  12  to  15  : 1 

10 

From  20  to  25  : 1 

Firebox  tubular 

12 

From  25  to  35  : 1 

Return  tubular 

15 

From  25  to  35  : 1 

Vertical  

15 

From  25  to  30  : 1 

Water  tube 

11 

From  35  to  40  : 1 

170 


BOILERS. 


The  above  rule  must  not  be  taken  as  furnishing  anything 
but  an  approximate  method,  since  the  same  boiler  will  give 
a different  horsepower  whenever  the  conditions  under  which 
it  is  operated  are  changed;  or,  in  other  words,  the  horsepower 
developed  depends  largely  on  the  amount  of  coal  burned  per 
square  foot  of  grate  area  per  hour,  the  velocity  and  character 
of  the  furnace  draft,  and  the  quality  of  the  coal  used.  In 
ordinary  practice,  however,  we  may  expect  an  evaporation 
of  from  8 to  11  lb.  of  water  from  and  at  212°  F.  for  each  pound 
of  good  coal  burned,  where  from  11  to  13  lb.  of  coal  are 
consumed  per  sq.  ft.  of  grate  surface  per  hr.,  or  about  from 
3 to  4 lb.  per  H.  P.  per  hr. 


CHIMNEYS. 

The  chimney  serves  the  double  purpose  of  creating  a draft 
and  carrying  away  obnoxious  gases.  The  production  of  the 
draft  depends  on  the  fact  that  the  furnace  gases  (the  products 
of  combustion)  passing  up  the  chimney  have  a high  tempera- 
ture, and  are,  consequently,  lighter  than  an  equal  volume  of 
outside  air  at  the  ordinary  temperature;  that  is,  the  pressure 
within  the  chimney  is  slightly  less  than  the  pressure  of  the 
outside  air.  Consequently,  the  air  will  flow  from  the  place  of 
higher  pressure  to  the  place  of  lower  pressure,  that  is,  into  the 
chimney  through  the  furnace. 

Suppose,  for  example,  the  average  temperature  of  the  gases 
in  a chimney  150  ft.  high  is  500°  F.  A pound  of  the  gases  at 
62°  F.  has  a volume  of  12.5  cu.  ft.;  its  volume  at  500°  is,  then, 

~2'-5  + 46°')  = 23  cu.  ft.  Therefore,  a column  of  the 

62  + 460 

150 

gases  1 ft.  square  and  150  ft.  long  would  weigh  — = 6.52  lb. 

150 

A similar  column  of  air  at  62°  F.  would  weigh  = 11.42  lb., 

nearly.  Hence,  the  pressure  of  the  draft  is  11.42  — 6.52  = 4.9 
lb.  per  sq.  ft.  = .941  in.  of  water.  It  is  evident  that  the  pres- 
sure of  the  draft  depends  on  the  temperature  of  the  furnace 
gases  and  the  height  of  the  chimney.  The  higher  the  chim- 
ney, the  lower  may  be  the  temperature  of  the  gases  to  produce 


CHIMNEYS. 


171 


the  same  draft,  and  the  greater  will  be  the  economy  of  the 
furnace.  In  general,  chimneys  are  not  built  much  less  than 
100  ft.  in  height. 

The  relation  between  the  height  of  the  chimney  and  the 
pressure  of  the  draft  in  inches  of  water  is  given  by  the  follow- 
ing formula:  rr/7.6  7.9\ 

P ~ H\  Ta  T')' 
where  p = draft  in  inches  of  water; 

H = height  of  chimney  in  feet; 

Ta  = absolute  temperature  of  outside  air; 

Tc  = absolute  temperature  of  chimney  gases. 

Absolute  temperatures  are  found  by  adding  460°  F.  to  the 
ordinary  temperatures. 

Example.— What  draft  pressure  will  be  produced  by  a 
chimney  120  ft.  high,  the  temperature  of  the  chimney  gases 
being  600°  F.  and  the  external  air  60°  F.? 

Solution.— By  the  formula  we  find 


7.6 


7.9 


j)  = -8 


in.  of 


V 460  + 60  460  + 600,/ 

water. 

The  draft  pressures  ordinarily  produced  by  chimneys  vary 
from  0 to  2 in.  of  water.  A water-gauge  pressure  of  1 in.  is 
equivalent  to  .03617  lb.  per  sq.  in.  Wood  requires  least  draft, 
and  the  small  sizes  of  anthracite  coal  the  greatest  draft.  To 
successfully  burn  anthracite,  slack,  or  culm,  a draft  of  1£  in. 
is  necessary. 

To  find  the  height  of  chimney  to  give  a specified  draft 
pressure,  the  formula  may  be  transformed: 

H = 


716_719* 

Ta  Tc 

Example.— Required  the  height  of  the  chimney  to  produce 
a draft  of  1£  in.  of  water,  the  temperature  of  the  gases  and  of 
the  external  air  being,  respectively,  550°  and  62°  F. 
Solution.— By  the  formula  we  find 
p _ 1.125 


II  = 


7^6 

T 

M a 


1A 

Te 


7.6 
522  ~ 


7.9 

1,010 


= 167  ft. 


The  sizes  of  chimneys  for  boilers  of  various  horsepowers 
are  given  in  the  following  table: 


172  BOILERS. 

Sizes  of  Chimneys  and  Horsepowers  of  Boilers. 


Height  of  Chimney  in  Feet. 

Actual  Area  in 
Sq.  Ft. 

Side  of  Sq.  in  In. 

Diameter  in  In.  | 

50 

60 

70 

80 

90 

100 

110 

125 

150 

175 

200 

Commercial  Horsepower. 

23 

25 

27 

1.77 

16 

18 

35 

38 

41 

2.41 

19 

21 

49 

54 

58 

62 

3.14 

22 

24 

65 

72 

78 

83 

3.98 

24 

27 

84 

92 

100 

107 

113 

4.91 

27 

30 

115 

125 

133 

141 

5.94 

30 

33 

141 

152 

163 

173 

182 

7.07 

32 

36 

183 

196 

208 

219 

8.30 

35 

39 

216 

231 

245 

258 

271 

9.62 

38 

42 

311 

330 

348 

365 

389 

12.57 

43 

48 

363 

4271 

449 

472 

503 

551 

15.90 

48 

54 

505 

539 

565 

593 

632 

692 

748 

19.64 

54 

60 

658 

694 

728 

776 

849 

918 

981 

23.76 

59 

66 

792 

835 

876 

934 

1,023 

1,105 

1,181 

28.27 

64 

72 

995 

1,038 

1,107 

1,212 

1,310 

1,400 

33.18 

70 

78 

1,163 

1,214 

1,294 

1,418 

1,531 

1,637 

38.48 

75 

84 

1,344 

1,415 

1,496 

1,639 

1,770 

1,893 

44.18 

80 

90 

1,537 

1,616 

1,720 

1,876 

2,027 

2,167 

50.27 

86 

96 

Example.— A round  chimney  100  ft.  high  is  to  he  used  for 
a battery  of  boilers  of  550  H.  P.  What  should  be  the  internal 
diameter? 

Solution.— Looking  under  column  100  in  “Height  of 
Chimney  in  Feet”  the  nearest  horsepower  is  565,  and  the 
diameter  corresponding  is  60  in.,  which  should  be  the  inter- 
nal diameter  of  the  chimney. 

Chimneys  are  usually  built  of  brick,  though  in  some  cases 
iron  stacks  are  preferred.  The  external  diameter  of  the  base 
should  be  TV  of  the  height,  in  order  to  provide  stability.  The 
taper  of  a chimney  is  from  ^ to  £ in.  to  the  foot  on  each  side. 
The  thickness  of  brickwork  is  usually  1 brick  (8  or  9 in.)  for 
25  ft.  from  the  top,  increasing  k brick  for  each  25  ft.  from  the 
top  downward.  If  the  inside  diameter  is  greater  than  5 ft., 
the  top  length  should  be  1£  bricks,  and  if  under  3 ft.,  it  may  be 


EXHAUST  HEATING. 


173 


& brick  in  thickness  for  the  first  10  ft.  A round  chimney  is 
better  than  a square  one,  and  a straight  flue  better  than  a 
tapering  one.  If  the  flue  is  tapering  the  area  for  calculation 
is  measured  at  the  top. 

The  flue  through  which  the  gases  pass  from  the  furnaces 
to  the  chimney  should  have  an  area  equal  to,  or  a little  larger 
than,  the  area  of  the  chimney.  Abrupt  turns  in  the  flue  or 
contractions  of  its  area  should  be  carefully  avoided,  as  they 
greatly  retard  the  flow  of  the  gases.  Where  one  chimney 
serves  several  boilers,  the  branch  flue  from  each  furnace  to 
the  main  flue  must  be  somewhat  larger  than  its  proportionate 
part  of  the  area  of  the  main  flue. 


SAFETY  VALVES. 

Balance  the  valve  and  lever  over  a sharp,  knife-like  edge, 
and  measure  the  distance  from  the  point  of  suspension  to  the 
fulcrum  (center  of  pin  on  which  the  lever  turns). 

Let  a = distance  thus  measured  in  inches; 

b = distance  from  center  of  valve  to  fulcrum  in  inches; 
x = distance  of  weight  from  fulcrum  in  inches; 

W = weight  in  pounds  hung  on  lever; 

Q = weight  of  lever  and  valve  in  pounds; 

A = area  of  safety  valve  in  square  inches; 
p = pressure  per  square  inch  in  the  boiler. 

Apb—Qa  JTr  Apb—Qa  Wx  + Qa 

Then,  x = - Tr.  -;  W = — - — ; p = 

W x * Ab 


EXHAUST  HEATING. 

Exhaust  steam  from  non-condensing  engines  usually  con- 
tains from  20$  to  25$  of  water  and  oil,  the  latter  being  employed 
to  lubricate  the  engine  cylinders.  Before  exhaust  steam  is 
allowed  to  enter  a heating  system,  the  water  and  oil  should  be 
separated  from  it. 

The  effect  of  turning  exhaust  steam  into  a heating  system 
is  to  form  a back  pressure  on  engine,  which  must  be  avoided 
as  far  as  possible  by  using  large  steam-distributing  pipes. 

A direct  connection  to  the  steam  boilers  through  a pressure- 
reducing  valve  must  be  employed,  to  automatically  furnish 


174 


BOILERS. 


steam  to  the  heating  system  when  the  exhaust  fails.  A relief 
valve,  also,  should  be  placed  upon  the  system,  so  that  surplus 
exhaust  steam  may  escape  to  the  atmosphere. 

To  proportion  an  exhaust-heating  system,  it  is  necessary  to 
know  about  how  many  square  feet  of  radiating  surface  we 
should  employ  to  properly  condense  the  exhaust  steam  from 
the  non-condensing  engines.  To  do  this  we  must  first  know 
the  weight  of  steam  that  would  be  discharged  from  the  engine. 


Class  of  Non-Condensing 
Engine. 


Water  Used  per 
Hour  for  Indicated 
Horsepower. 


Compound  automatic 

Simple  Corliss 

Simple  automatic 

Simple  throttling 


25  lb. 
30  lb. 
35  lb. 
40  lb. 


From  this  must  be  deducted  about  10$  for  condensation  in 
the  cylinders,  etc.,  in  order  to  obtain  the  real  available  weight 
of  steam  for  heating  purposes. 

Approximate  Ratio  Between  Cubic  Contents  and  Radi- 
ator Surface  for  Exhaust  Heating. 


Class  of  Building. 

Direct 

Radiation. 

Indirect 

Radiation. 

Blower 

System. 

Dwellings  

sq.ft,  cu.ft. 

1 to  50 
1 to  70 
1 to  100 
1 to  200 

sq.ft,  cu.ft. 

1 to  40 
1 to  60 
1 to  80 
1 to  150 

sq.ft,  cu.ft. 

1 to  300 
1 to  365 
1 to  500 
1 to  900 

Offices  

Stores  and  shops 

Churches,  etc 

The  figures  in  the  foregoing  tables  simply  form  a reason- 
able average,  and  allowance  must  be  made  for  exposure,  etc. 

Each  square  foot  of  direct  radiating  surface  gives  off  to  the 
air  around  it  about  1£  thermal  units  per  hour  per  degree  of 
difference  between  the  temperature  of  the  steam  and  that  of 
the  surrounding  air.  This  is  equivalent  to  about  £ lb.  of 
steam  per  hr.,  or,  in  other  words,  about  4 to  4£  sq.  ft.  of  surface 
to  each  pound  of  steam  to  be  condensed. 


BLUEPRINTS. 


175 


MACHINE  DESIGN. 


BLUEPRINTS. 

Blueprint  paper  for  copying  tracings  of  plans  and  other 
drawings  may  be  prepared  as  follows:  Dissolve  1 oz.,  avoir- 
dupois, of  ammonia  citrate  of  iron  in  6 oz.  of  water,  and  in  a 
separate  bottle  dissolve  the  same  quantity  of  potassium  ferri- 
cyanide  in  6 oz.  of  water.  Keep  these  solutions  separate,  and 
in  a dark  place,  or  in  opaque  bottles. 

To  prepare  the  paper,  mix  equal  quantities  of  the  two 
solutions,  and  with  a sponge  spread  it  evenly  over  the  sur- 
face. Let  the  paper  remain  in  a horizontal  position  until 
the  chemical  has  set  on  the  surface,  which  will  take  but  a 
few  minutes;  then  hang  the  paper  up  to  dry.  In  preparing 
the  paper  darken  the  room  by  pulling  down  the  shades,  as 
direct  rays  of  light  aifect  sensitized  surfaces.  The  prepared 
paper  should  be  kept  in  a closed  drawer,  well  covered  with 
heavy  paper,  so  that  no  light  can  come  in  contact  with  the 
sensitized  surface;  otherwise  it  will  lose  much  of  its  value. 

To  make  a blueprint  from  a tracing,  lay  the  tracing  with  ink 
side  down  against  the  glass  of  the  printing  frame,  then  take 
the  prepared  paper,  and  place  the  sensitized  surface  down  on 
the  tracing.  On  the  top  of  the  paper  place  the  felt  cushion, 
on  top  of  which  place  the  hinged  back  of  the  printing 
frame,  after  which  expose  to  the  sunlight.  The  exposure 
will  vary  in  sunlight  from  about  3 to  10  minutes.  After  the 
exposure,  wash  the  paper  thoroughly  in  a trough  of  cold 
water  for  about  10  minutes,  and  hang  it  up  to  dry. 

The  print  after  washing  should  be  of  a deep-blue  color, 
with  clear  white  lines.  If  the  color  is  a pale  blue,  this  indi- 
cates that  the  print  has  not  had  sufficient  exposure,  and  if 
the  lines  of  the  drawing  are  not  perfectly  clear  and  white, 
that  the  exposure  has  been  too  long. 

Corrections  may  be  made  on  the  print  with  an  ordinary 
writing  or  ruling  pen  and  a solution  of  washing  soda,  caustic" 
potash,  strong  ammonia,  or  any  other  alkali.  When  any  of 
these  are  mixed  with  carmine  ink,  the  marks  on  the  print 
will  be  red,  thus  making  the  corrections  clear. 


176 


MACHINE  DESIGN. 


MACHINE  TOOLS. 


SPEED  OF  EMERY  WHEELS. 

The  speed  most  strongly  recommended  by  their  manufac- 
turers is  a peripheral  velocity  of  5,500  ft.  per  min.  for  all  sizes. 
All  things  being  considered,  it  is  stated  that  no  advantage 
is  gained  by  exceeding  this  speed.  If  run  much  slower 
than  this,  the  wear  on  the  wheels  is  much  greater  in  pro- 
portion to  the  work  accomplished,  and  if  run  much  faster, 
the  wheel  is  likely  to  burst. 

SPEED  OF  GRINDSTONES. 

Grindstones  used  for  grinding  machinists’  tools  are  usu- 
ally run  so  as  to  have  a peripheral  speed  of  about  900  ft.  per 
min.,  and  those  used  for  grinding  carpenters’  tools  at  about 
600  ft.  per  min.  With  regard  to  safety,  it  may  be  stated  in 
general  that  with  any  size  of  grindstone  having  a compact 
and  strong  grain,  a peripheral  velocity  of  2,800  ft.  per  min. 
should  not  be  exceeded.  . 

SPEED  OF  POLISHING  WHEELS. 

Polishing  wheels  are  run  at  about  the  following  peripheral 


speeds: 

Leather-covered  wooden  wheels 7,000  ft.  per  min. 

Walrus-hide  wheels. 8,000  ft.  per  min. 

Rag  wheels 7,000  ft.  per  min. 


SPEED  OF  CUTS  FOR  MACHINE  TOOLS. 

Brass:  Use  high  speeds,  about  the  same  as  for  wood. 

Bronze : 6 to  18  ft.  per  min.,  according  to  alloy  used. 

Cast  or  wrought  iron:  20  ft.  per  min.  is  a good  average  for 
all  machines,  except  millers.  30  is  about  the  maximum. 

Machinery  steel:  15  ft.  on  shapers,  planers,  and  slotters. 
20  to  45  on  turret  lathes,  according  to  cut. 

Tool  steel:  8 to  10  ft. 

Milling  Cutters.— Gun  metal , 80  ft.  per  min.;  cast  iron , 30; 
wrought  iron , 35  to  40;  machinery  steel , 30.  These  are  good 
speeds  to  adopt,  with  a view  to  economy,  time  required  for 
regrinding,  etc. 


MACHINE  TOOLS. 


177 


Twist  Drills.— The  best  results  are  obtained  when  the  rates 
of  speed  of  twist  drills  are  as  given  in  the  following  table: 


Revolutions  of  Drills  per  Minute. 


of  Drills. 

Steel. 

Iron. 

Brass. 

Ps 

940 

1,280 

1,560 

460 

660 

785 

8 

310 

420 

540 

230 

320 

400 

Ps 

190 

260 

320 

150 

220 

260 

p 

130 

185 

230 

115 

160 

200 

Ps 

100 

140 

180 

95 

130 

160 

I 

a 

85 

115 

145 

75 

105 

130 

70 

100 

120 

Vs 

65 

90 

115 

It 

62 

85 

110 

1 

58 

80 

100 

IPs 

54 

75 

95 

52 

70 

90 

i % 

49 

66 

85 

46 

62 

80 

IPs 

44 

60 

75 

42 

58 

72 

Ip 

40 

56 

69 

39 

54 

66 

IPs 

37 

51 

63 

36 

49 

60 

ill 

34 

47 

58 

33 

45 

56 

ill 

32 

43 

54 

31 

41 

52 

111 

30 

40 

51 

2 

29 

39 

49 

The  following  are  recommended  as  the  best  rates  of  feed 
for  twist  drills: 


Diameter  of  drill  in 

inches 

Number  of  revolu- 

K 

% K 

% 1 IK 

tions  per  inch  depth 
of  hole 

125 

125 

120  to  140 

1 in.  feed  per  min. 

178 


MACHINE  DESIGN. 


CHANGE  GEARS  REQUIRED  FOR  CUTTING  SCREW 
THREADS. 

The  pitch  of  a single-threaded  screw  is  the  distance 
between  two  adjacent  threads,  measured  on  a line  parallel 
to  the  axis  of  the  screw;  or,  in  any  screw,  whether  single-  or 
multiple-threaded,  it  is  the  distance  the  nut  is  moved  by  1 
revolution  of  the  screw.  Usually,  a screw  is  spoken  of  as 
having  a certain  number  of  threads  to  the  inch,  and  this  is 
equal  to  the  number  of  revolutions  the  screw  must  make  in 
order  to  move  the  nut  a distance  of  1 inch;  so,  whether  the 
screw  is  single-  or  multiple-threaded,  the  pitch  is  always 
equal  to  1 divided  by  the  number  of  revolutions  that  the 
screw  must  make  in  order  to  move  the  nut  1 inch. 

The  Simple-Geared  Lathe.— In  Fig.  1 is  shown  the  usual 
arrangement  of  the  change  gears  of  a simple-geared  screw- 
cutting lathe.  By  a simple-geared  lathe  is  meant  a lathe  in 


which  the  change  gears  are  so  arranged  that  the  circum- 
ferential velocity  of  the  change  gear  on  the  stud  is  the 
same  as  that  of  the  change  gear  on  the  lead  screw,  which 
means  that,  when  the  change  gear  on  the  stud  has  rotated, 
say,  5 teeth,  the  change  gear  on  the  lead  screw  has  also 
rotated  5 teeth,  whatever  the  diameter  of  these  gears,  or 
of  any  intermediate  gears  between  them,  may  be. 

Referring  to  Fig.  1,  the  gear  a is  fastened  to  the  spindle  b 
and  drives  another  gear  c by  means  of  either  one  of  the 


MACHINE  TOOLS. 


179 


reversing  gears  d,  d' . The  gear  c is  keyed  to  one  end  of  the 
spindle  e;  this  spindle  is  called  the  stud , and  carries  on  its 
outer  end  a change  gear  /.  The  lead  screw  g carries  a 
change  gear  h\  and  these  two  change  gears  / and  h are 
connected  by  means  of  the  idler  gear  i,  so  that  gear/ drives 
gear  h,  and  with  it,  the  lead  screw  g. 

In  making  calculations  for  the  change  gears  of  a simple- 
geared  screw-cutting  lathe,  the  idler  gear  i is  ignored,  as  it 
is  only  introduced  to  connect  gears  / and  h.  The  gears 
d and  d'  are  also  ignored,  since  they  are  only  used  to  change 
the  direction  of  rotation  of  the  gear  c,  their  duty  being  to 
facilitate  the  cutting  of  either  right-hand  or  left-hand 
threads;  when  d meshes  with  gear  a , as  shown  in  Fig.  1,  a 
a right-hand  thread  is  cut,  and  when  df  meshes  with  gear  o, 
a left-hand  thread  is  cut. 

The  number  of  teeth  in  the  gear  a is  not  always  the  same 
as  the  number  of  teeth  in  the  gear  c;  it  is  so  in  some  lathes, 
but  in  others  it  is  not;  hence,  in  calculating  the  change 
gears  for  any  lathe,  the  number  of  teeth  in  the  gears  a and  c 
must  be  taken  into  account. 

By  the  following  formulas  and  rules,  the  number  of  teeth 
required  in  each  change  gear  in  order  to  cut  a given  number 
of  threads  to  the  inch,  or  the  number  of  threads  to  the  inch 
that  given  change  gears  will  produce  may  be  found. 

Let  a = number  of  teeth  in  the  spindle  gear  a ; 
c = number  of  teeth  in  the  gear  c; 

/ — number  of  teeth  in  the  change  gear  on  stud; 
h = number  of  teeth  in  the  change  gear  on  lead 
screw; 

g = number  of  threads  to  the  inch  in  the  lead  screw; 
n = number  of  threads  to  the  inch  to  be  cut. 


Then,  n = 

_ gch 

a/  ' 

(1) 

h = ^f. 

gc 

(3) 

n a 

(2) 

f _ &ch 

J na' 

(4) 

Now,  of  the  gears  /*,/,  c,  a,  a and /are  the  drivers , and  c and 
h being  driven  by  a and  /,  are  called  the  driven  gears; 
remembering  this,  we  deduce,  from  formula  (1),  the  following 
rule  for  simple-geared  screw-cutting  lathes: 


180 


MACHINE  DESIGN. 


Rule.— The  number  of  threads  to  the  inch  to  be  cut  is  equal  to 
the  number  of  threads  to  the  inch  in  the  lead  screw , multiplied  by 
the  product  of  the  number  of  teeth  in  each  driven  gear , and 
divided  by  the  product  of  the  number  of  teeth  in  each  driving  gear. 

Example.— If  the  lead  screw  g of  a simple-geared  lathe  has 
5 threads  to  the  inch,  and  the  gear  a has  21  teeth,  the  gear  c 
42  teeth,  the  change  gear  / 60  teeth,  and  the  change  gear  h 
72  teeth,  how  many  threads  to  the  inch  will  be  cut? 

Solution. — Using  formula  (1),  we  have 


gch  __  5 X 42  X 72 


= 12  teeth. 


af  “ 21X60 

From  formula  (2)  we  deduce  the  following  rule  for  simple- 
geared  screw-cutting  lathes: 

Rule. — The  number  of  teeth  in  the  change  gear  on  the  lead 
screw,  divided  by  the  number  of  teeth  in  the  change  gear  on  the 
stud,  is  equal  to  the  product  of  the  number  of  threads  to  the  inch 
to  be  cut  and  the  number  of  teeth  in  the  driving  spindle  gear , 
divided  by  the  product  of  the  number  of  threads  to  the  inch  in 
lead  screw  and  the  number  of  teeth  in  the  fixed  gear  on  the  stud. 

Example.— If  the  lead  screw  g of  a simple-geared  lathe 
has  8 threads  to  the  inch,  and  the  gear  a has  16  teeth,  and  the 
gear  c 32  teeth,  how  many  teeth  must  there  be  in  each  of 
the  gears  / and  h in  order  that  the  lathe  may  cut  10  threads 
to  the  inch? 

Solution.— Using  formula  (2), 

h na  10  X 16  5 

f ~ gc~  8X32  ~ 8’ 

and,  if  it  were  possible  to  have  gears  with  5 and  8 teeth, 
respectively,  then  a solution  of  the  problem  would  be, 
h = 5,/  = 8.  It  is  evident  that  such  gears  are  impracticable; 
but,  as  it  does  not  change  the  value  of  a fraction  to  multiply 
both  numerator  and  denominator  by  the  same  number,  we 
may  multiply  5 and  8,  each  by  such  a number  that  the  result- 
ing numbers  of  teeth  in  the  gears  are  satisfactory.  There  is 
evidently,  therefore,  more  than  one  solution  to  the  problem— 
for  if  we  multiply  by  10  we,  shall  have  h = 50,  / = 80,  which 
would  give  12  threads  to  the  inch;  and  if  we  multiply  by  13, 
we  shall  have,  as  another  solution,  h = 65,  / = 104,  which 
would  also  give  12  threads  to  the  inch,  because  fife  = |. 


MACHINE  TOOLS. 


181 


Having  found  that  j = |,  it  is  customary  in  practice  to 

choose  the  change  gears  in  the  following  manner:  From  the 
assortment  of  gears  belonging  to  the  lathe,  choose  one  of 
convenient  diameter,  the  number  of  whose  teeth  is  divisible 
by  either  the  numerator  5 or  the  denominator  8,  and,  after 
dividing  by  one  of  these  numbers,  multiply  both  numerator 
and  denominator  by  the  quotient. 


Example.— Given,  - = •§,  to  find  the  number  of  teeth  in 

the  two  change  gears  h and/,  respectively. 

Solution.— Choose  a gear  of  convenient  diameter,  the 
number  of  whose  teeth,  say  60,  is  divisible  by  either  5 or  8,  in 
this  case  by  5;  divide  60  by  5,  and  the  answer  is  12.  Then,  • 


5X12  = 60 
8 X 12  96’ 

that  is,  h has  60  teeth,  and  / 96  teeth. 

If  one  of  the  change  gears  is  given,  and  it  is  desired  to  find 
the  number  of  teeth  in  the  other  change  gear  in  order  to  cut 
a given  number  of  threads  to  the  inch,  use  either  formula  (3) 
or  formula  (4)  according  as  the  number  of  teeth  in  gear  h or 
in  gear /is  required.  After  the  examples  given,  these  formu- 
las will  not  need  explanation. 

In  a simple-geared  screw-cutting  lathe,  it  is  often  possible 
to  cut  a fractional  number  of  threads  to  the  inch,  as  is  the  case 
in  the  following  example: 

Example. — If  the  lead  screw  g has  2 threads  per  inch,  and 
the  gear  a has  20  teeth,  and  the  gear  c has  20  teeth,  how  many 
teeth  must  there  be  in  each  of  the  change  gears  / and  h , in 
order  to  cut  5£  threads  to  the  inch  ? 

Solution.— Using  formula  (2), 

h na  5i  X 20  5£ 

7 “ Jc~  2X20'“  IT* 

Then,  choosing  a gear  whose  number  of  teeth,  say  32,  is 
divisible  by  2,  divide  32  by  2 and  the  quotient  is  16.  Then, 

Slvifi  «4 

*-  — = j that  is,  h has  84  teeth,  and  / 32  teeth.  In 

many  cases,  however,  it  is  impossible,  out  of  the  assortment 
of  gears  supplied  with  a simple-geared  screw-cutting  lathe,  to 


182 


MACHINE  DESIGN. 


find  gears  to  cut  a screw  of  the  required  number  of  threads 
to  the  inch.  In  such  cases,  it  becomes  necessary  either  to 
make  suitable  gears  or  to  resort  to  a compound-geared  lathe. 

The  Compound-Geared  Lathe. — In  Fig.  2 is  shown  the  usual 
arrangement  of  the  change  gears  of  a compound-geared 
screw-cutting  lathe.  The  difference  between  this  and  the 
simple-geared  lathe  lies  in  putting  two  change  gears  of  differ- 
ent sizes  on  one  spindle,  in  place  of  the  idler  between  the 
gear  on  the  stud  and  the  gear  on  the  lead  screw.  These  two 
gears  on  one  spindle  are  shown  at  i and  j in  Fig.  2,  gear  j 
meshing  with  gear  h on  the  lead  screw,  and  gear  i meshing 
with  gear/ on  the  stud. 


From  the  following  formulas,  the  number  of  teeth  in  each 
change  gear,  or  the  number  of  threads  per  inch  that  can  be 
cut  with  given  change  gears,  can  be  found. 

Let  a = number  of  teeth  in  the  spindle  gear  a; 
c = number  of  teeth  in  the  gear  c; 

/ = number  of  teeth  in  the  change  gear/; 
h = number  of  teeth  in  the  change  gear  h ; 
i ==  number  of  teeth  in  the  change  gear  i , which 
meshes  with  the  change  gear/; 
j = number  of  teeth  in  the  change  gear  j,  which 
meshes  with  the  change  gear  h ; 
g = number  of  threads  to  the  inch  in  the  lead  screw; 
n = number  of  threads  to  the  inch  to  be  cut. 

gXchi  /rx 
Then,  n = --jt-.  (o) 


MACHINE  TOOLS. 


183 


Now,  remembering  that  gears  a,  f and  j are  the  drivers, 
and  gears  c,  h,  and  i are  the  driven  gears,  and  also  that  the 
idlers  are  ignored  in  all  calculations,  we  can,  from  formula 
(5),  deduce  the  following  rule  for  compound-geared  screw- 
cutting lathes: 

Rule. — The  number  of  threads  to  the  inch  to  be  cut  is  equal  to 
the  number  of  threads  to  the  inch  in  the  lead  screw , multiplied  by 
the  product  of  the  number  of  the  teeth  in  each  of  the  driven  gears , 
and  divided  by  the  product  of  the  number  of  teeth  in  each  of  the 
dnving  gears. 

Example.— If  the  lead  screw  (7  of  a compound-geared  lathe 
has  2 threads  to  the  inch,  and  the  gear  a has  20  teeth,  gear  c 40 
teeth,  change  gear  / 48  teeth,  change  gear  i 72  teeth,  change 
gear  j 36  teeth,  and  change  gear  h 96  teeth,  how  many  threads 
to  the  inch  will  be  cut  ? 

Solution— Using  formula  (5),  we  have 


gX  chi  _ 2X40X96X72 


afj 


= -20  X 48  X 36"  = 16  threadS  t0  the  iQCh- 


If  it  is  desired  to  find  what  combination  of  change  gears 
will  enable  us  to  cut  a given  number  of  threads  to  the  inch, 
the  following  formula  may  be  used: 


(6) 


i_  _ naf 

J ~ gch' 

From  this  formula  the  following  rule  is  deduced: 

Rule. — Of  the  change  gears  of  a lathe , any  driven  gear  divided 
by  any  driver  gear  is  equal  to  the  product  of  the  numbers  of  teeth 
in  each  of  the  other  driver  gears  and  the  number  of  threads  to  the 
inch  to  be  cut , divided  by  the  product  of  the  numbers  of  teeth  in 
each  of  the  other  driven  gears  and  the  number  of  threads  to  the 
inch  in  the  lead  screw. 

Example.— In  a compound-geared  lathe,  in  which  the  lead 
screw  has  5 threads  to  the  inch,  gear  a 20  teeth,  gear  c 40 
teeth,  and  the  number  of  threads  per  inch  to  be  cut  is  3i, 
what  must  be  the  number  of  teeth  in  each  of  the  change 
gears  ht  itj,f? 

Solution.— Using  formula  (6),  we  have 


i _ naf 
j ~ g~cK 


184 


MACHINE  DESIGN. 


From  the  assortment  of  gears  belonging  to  the  lathe,  choose, 
for  the  driven  gear  h,  one  whose  number  of  teeth,  say  28,  can 
be  divided  by  the  number  of  threads  per  inch  to  be  cut,  in 
this  case  3£;  28  is  a multiple  of  3£,  because  it  is  obtained  by 
multiplying  3£  by  8.  Substitute  this  value  in  place  of  h;  then 
choose  any  gear  of  convenient  size,  say  one  having  40  teeth, 
and  substitute  40  in  place  of  /;  we  shall  then  have, 
i _ n a X 40. 
j ” g c X 28’ 

or,  substituting  the  given  values  of  n,  a,  g,  and  c, 
i _ 3£  X 20  X 40  1 

j 5X40X  28  “2* 

Choose,  for  j,  a gear  whose  number  of  teeth,  say  60,  is 
divisible  by  2;  then,  dividing  the  number  of  teeth  in,;  by  2,  we 
have  60  2 = 30.  Now  multiplying  both  terms  of  the  frac- 

tion £ by  30, 

£ _ 1 X 30  _ 30, 
j ~ 2 X 30  “ 60  ’ 

that  is,  i — 30,  and  j = 60.  Hence,  one  solution  of  the  prob- 
lem is,  h = 28;  i = 30;  j = 60;  / = 40. 


HORSEPOWER  OF  ENGINES,  BOILERS, 
AND  PUMPS. 

THEORETICAL  HORSEPOWER. 

The  theoretical  horsepower  of  any  machine  that  uses  a 
fluid  (steam,  gas,  water,  etc.)  as  a motive  power,  or  that  dis- 
charges a fluid  (i.  e.,  a pump  or  a fan),  may  be  readily  com- 
puted by  the  following  formula,  in  which  v is  the  volume  of 
the  fluid  used  or  discharged  in  cubic  feet  per  minute,  and  p 
is  the  average  pressure  in  pounds  per  square  inch: 
tt  p _ 144  vv 
' ' 33,000 ' 

If,  in  the  above  formula,  allowance  for  friction,  etc.  is 
made,  the  final  result  will  be  the  actual  horsepower. 

Example.— A ventilating  fan  delivers  5,000  cu.  ft.  of  air 
per  min.  at  a pressure  of  .56  lb.  above  the  atmospheric  pressure; 
what  is  the  theoretical  horsepower  required  to  drive  the  fan  ? 


HORSEPOWER. 


185 


Solution.— 
H.  P.  = 


144  v p 
33,000 


144  X 5,000  X .56 


= 12.218. 


33,000 

If  all  hurtful  resistances  are  taken  in  this  case  as  20$  of  the 
total  horsepower,  the  actual  horsepower  will  be 


12.218  (1  — .20)  = 12.218  -r-  .80  = 15.27  H.  P. 

Example.— The  mean  effective  pressure  computed  from  an 
indicator  card  taken  from  the  air  cylinder  of  an  air  com- 
pressor is  30.6  lb.  per  sq.  in.;  diameter  of  cylinder,  28  in.; 
stroke,  48  in.;  number  of  strokes  per  minute,  108;  what  is  the 
horsepower? 

Solution.— In  this  case 


v 

Hence, 
144  vp  __ 
3p00  — 


282  X .7854  X 48  X 108 
1,728 


cu.  ft.  per  min. 


144  X 282  X .7854  X 48  X 108  X 30.6 
1,728  X 33,000 


246.66  H.  P. 


HORSEPOWER  OF  AN  ENGINE. 

Let  P = mean  effective  pressure  in  pounds  per  square 
inch  on  the  piston  during  one  stroke; 

L = length  of  stroke  in  feet; 

A = area  of  piston  in  square  inches; 

N = number  of  strokes  per  minute; 

D — diameter  of  piston  in  inches. 

Then,  to  find  the  indicated  horsepower, 

PLAN  _ 238  P L D2  N 
* * * 33,000  ~ 10,000,000  ’ 

The  actual  horsepower  may  be  taken  as  three-fourths  of  the 
indicated  horsepower.  The  mean  effective  pressure  may  be 
found  exactly  by  taking  some  indicator  cards,  finding  the 
areas  by  means  of  a planimeter,  and  dividing  the  area  by 
the  length  of  the  card.  Multiply  the  result  by  the  scale  of  the 
indicator  spring,  and  the  product  will  be  the  mean  effective 
pressure,  or  M.  E.  P.  If  no  planimeter  is  at  hand,  divide  the 
card  into  10  equal  parts  and  measure  each  part  in  the  middle, 
as  shown  by  the  dotted  lines  in  the  following  figure. 

Add,  all  the  dotted  ordinates  together,  and  divide  by  10; 
this  result,  multiplied  by  the  scale  of  the  indicator  spring, 
gives  the  M.  E.  P. 


186 


MACHINE  DESIGN. 


Thus,  suppose  a double-acting  engine  26"  X 30",  making 
80  rev.  per  min.  (80  R.  P.  M.),  gives  an  indicator  card  that, 
being  divided  up  as  shown  in  the  figure  and  measured,  gives, 
for  the  total  length  of  the  ordinates,  21.4  in.  This  divided  by 


10  = 2.14  in.  for  the  length  of  the  mean  ordinate.  If  a No.  40 
spring  is  used  in  the  indicator,  every  inch  measured  ver- 
tically on  the  diagram  = 40  lb.  per  sq.  in.,  and  2.14  X 40  = 
85.6  lb.  per  sq.  in.  for  the  M.  E.  P.  on  the  piston.  Then  the 
indicated  horsepower,  or  I.  H.  P.,  equals 

PLAN  85.6  X X (.7854  X 262)  X (2X80)  _ 

83,000  33,000  U,88‘ 


The  calculation  is  rendered  much  easier  by  using  the  sec- 
ond formula.  Thus, 

T tt  t>  238  X 85.6  X f § X 262  X (2  X 80)  ecn  00 

L - 16,000,600  “ 550-88' 

If  an  indicator  card  cannot  be  obtained,  a fair  approxima- 
tion to  the  M.  E.  P.  may  be  obtained  by  adding  14.7  to  the 
gauge  pressure,  and  multiplying  the  number  opposite  the 
fraction  indicating  the  point  of  cut-off  in  the  following  table 
by  the  boiler  pressure.  Subtract  17  from  the  product,  and 
multiply  by  .9.  The  result  is  the  M.  E.  P.  for  good  simple 
non-condensing  engines.  If  the  engine  is  a simple  con- 
densing engine,  subtract  the  pressure  in  the  condenser  instead 
of  17.  The  fraction  indicating  the  point  of  cut-off  is  obtained 
by  dividing  the  distance  that  the  piston  has  traveled  when 
the  steam  is  cut  off  by  the  whole  length  of  the  stroke.  Thus, 
if  the  stroke  is  30  in.,  and  the  steam  is  cut  off  when  the  piston 


THE  SLIDE  VALVE. 


187 


has  traveled  20  in.,  the  engine  cuts  off  at  = | stroke.  For 
a f cut-off,  and  92-lb.  gauge  pressure  in  the  boiler,  the  M.  E.  P. 
is  [(92  + 14.7)  X .917  — 17]  X .9  = 72.76  lb.  per  sq.  in. 


Cut-off. 

Constant. 

Cut-off. 

Constant. 

Cut-off. 

Constant. 

% 

.566 

% 

.771 

% 

.917 

* 

.603 

.4 

.789 

.7 

.926 

.659 

A 

.847 

% 

.937 

.3 

.708 

.6 

.895 

.8 

.944 

Yz 

.743 

% 

.904 

Vs 

.951 

THE  SLIDE  VALVE. 

Figs.  A,  B,  C , and  D show  sections  of  an  ordinary  D slide 
valve  at  different  points  of  its  travel.  Fig.  A shows  the  valve 
in  its  central  position,  with  the  center  of  the  valve  in  line 
with  the  center  line  of  the  exhaust  port.  The  names  of  the 
various  parts  are  as  follows:  p and  p are  the  steam  ports;  e is 
the  exhaust  port;  s,  s is  the  naive  seat;  the  amount  o by  which 
the  valve  overlaps  the  outer  edges  of  the  steam  ports  is  the 
outside  lap;  the  amount  i by  which  the  valve  overlaps  the 
inside  edges  of  the  steam  port  is  called  the  inside  lap;  the 
amount  l (Fig.  C)  that  the  port  is  open  when  the  piston  is  at 
the  end  of  the  stroke  is  called  the  lead.  The  valve  travel  is  the 
total  distance  in  one  direction  that  the  valve  can  be  moved 
by  the  eccentric;  it  is  the  total  distance  between  two  extreme 
positions  of  the  valve.  The  displacement  of  the  valve  is  the 
distance  that  the  valve  has  moved  (in  either  direction)  from 
its  central  position. 

The  line  joining  the  center  of  the  eccentric  with  the  center 
of  the  crank-shaft  is  called  the  eccentric  radius.  When  the 
eccentric  radius  makes  a right  angle  with  the  center  line  of 
the  crank,  that  is,  when  the  eccentric  radius  is  vertical  (see 
oe,  Fig.  E),  the  valve  is  in  its  central  position,  provided  the 
valve  seat  is  horizontal,  as  is  usually  the  case.  When  the 
crank  is  on  a dead  center,  say  a,  Fig.  E , the  valve  must  be  in 
the  position  shown  in  Fig.  C;  that  is  to  say,  the  crank  must 


188 


MACHINE  DESIGN. 


have  moved  from  its  central  position  an  amount  equal  to  the 
outside  lap  plus  the  lead.  In  order  that  this  may  happen, 
the  eccentric  must  be  at  c,  Fig.  E.  The  angle  eoc,  through 
which  the  eccentric  must  be  moved  from  its  vertical  position 
when  the  crank  is  on  a dead  center,  is  called  the  angle  of 
advance. 


THE  SLIDE  VALVE. 


189 


In  Fig.  B,  the  valve  is  shown  in  its  extreme  position  at  the 
right.  The  distance  marked  m is  the  maximum  port  opening. 
It  matters  not  whether  the  outer  edge  of  the  valve  travels 
beyond  the  inner  edge  of  the  port  or  falls  short  of  it,  as  in  the 
figure,  the  distance  m between  the  edge  of  the  valve  and  the 
edge  of  the  port  when  the  valve  is  in  its  extreme  position  is 
the  maximum  port  opening.  If,  in  Fig.  C,  the  valve  were 
shown  moving  to  the  left,  a little  farther  movement  would 
bring  the  left  outer  edge  just  even  with  the  outer  edge  of  the 
left  steam  port,  and  from  here  on  to  the  end  of  the  stroke  no 
more  steam  could  enter  the  left  end  of  the  cylinder;  in  other 
words,  the  valve  cuts  off  at  this  point.  A little  farther  move- 
ment of  the  valve  to  the  left  brings  the  valve  to  the  position 
shown  in  Fig.  D,  with  the  right  inner  edge  opposite  the  inner 
edge  of  the  right  steam  port;  it  is  at  this  point  that  compres- 
sion begins. 

When  designing  a valve  for  an  engine,  some  of  the  above 
quantities  are  assumed  and  the  remaining  ones  are  required; 
these  may  be  found  by  means  of  the  diagram  shown  in  Fig.  E. 

Let  a b,  Fig.  E,  drawn  to  any  convenient  scale,  represent 
the  stroke  of  the  engine;  then  a db  will  represent  the  crank- 
pin  circle.  About  o,  the  center  of  the  crankpin  circle, 
describe  a circle  a' eb whose  diameter  a'b'  is  equal  to  the 
actual  travel  of  the  valve.  Draw  the  line  gh  parallel  to  ab 
and  at  a distance  from  it  equal  to  the  lead  of  the  valve. 
Then,  with  a radius  o' j equal  to  the  outside  lap  of  the  valve, 
describe  a circle,  called  the  outside  lap  circle , tangent  to  the 
line  gh,  and  having  its  center  o'  on  the  circle  a'eb'.  Draw 
the  line  oo',  and  produce  it  to/;  then  fob  = eoc  = angle  of 
advance. 

Now,  draw  any  position  of  the  crank  center  line,  such  as 
a o,  and  drop  upon  it,  from  the  point  o',  a perpendicular;  the 
length  of  this  perpendicular  (marked  r in  Fig.  E)  is  the  dis- 
placement of  the  valve  for  that  position  of  crank  center  line. 

About  the  center  o'  with  a radius  equal  to  the  inside  lap  of 
the  valve,  describe  a circle;  this  is  called  the  inside  lap  circle. 

The  radius  od,  drawn  from  the  point  o tangent  to  the 
outside  lap  circle,  is  the  position  of  the  center  line  of  crank 
at  the  point  of  cut-off.  Drop  a perpendicular  from  point  d , 


190 


MACHINE  DESIGN. 


meeting  the  line  ab  at  then  ak  is  the  distance  moved  by 
piston  before  cut-off,  and  the  fraction  of  the  stroke  at  which 

cl  k 

the  valve  cuts  off  is  represented  by  the  fraction  — . 

Draw  the  radius  o l tangent  to  the  upper  side  of  the  inside 
lap  circle,  and  it  will  be  the  position  of  the  center  line  of  the 
crank  when  compression  commences;  if  a perpendicular  is 
dropped  from  point  l , meeting  the  line  ab  at p,  the  fraction 
of  the  stroke  of  piston  at  which  compression  begins  will  be 

represented  by  the  fraction 

In  like  manner,  the  radius  ora,  drawn  tangent  to  the 
lower  side  of  the  inside  lap  circle,  is  the  position  of  the  center 

line  of  the  crank  at  the  moment  of  release;  and  — ? is  the 

a b 

fractional  part  of  the  stroke  at  which  the  expanding  steam 
is  released. 

The  maximum  steam-port  opening  is  equal  to  on,  n being 
the  point  of  intersection  of  the  outside  lap  circle  with  the 
angle  of  advance  line  o /. 

The  essential  features  of  the  valve  diagram  having  been 
given,  the  following  examples  will  make  clear  its  application 
in  practice: 

Example  1. — Given,  the  point  of  cut-off,  the  point  of 
release,  the  lead,  and  the  maximum  port  opening,  to  find  the 
valve  travel,  the  outside  and  inside  lap,  the  angle  of 
advance,  and  the  point  of  compression. 

Solution. — Draw  to  a convenient  scale  the  crankpin 
circle  ad b,  Fig.  E,  having  its  center  at  o , and  its  diameter  ab 
equal  to  the  stroke  of  the  piston. 

From  the  point  a,  lay  off,  on  the  line  a b,  the  distances  a k 

and  ay,  so  that  and  are  equal,  respectively,  to  the 

fractions  of  the  stroke  at  which  cut-off  and  release  are  to 
occur.  At  k and  y draw  perpendiculars  to  the  line  a b,  inter- 
secting the  crankpin  circle  at  d and  ra,  respectively;  the 
radii  o d and  ora  will  represent  the  positions  of  the  crank  at 
cut-off  and  release,  respectively.  Now  draw  gh  parallel  to  ab, 
and  at  a distance  above  it  equal  to  the  lead;  then,  about  o as 


THE  SLIDE  VALVE. 


191 


a center,  and  with  a radius  equal  to  the  given  maximum 
port  opening,  describe  an  arc.  Find  by  trial  a center  o', 
from  which  a circle  can  be  drawn  tangent  to  this  arc,  and 
also  to  the  radius  o d,  and  to  the  line  g h.  The  radius  of  this 
circle  will  be  the  required  outside  lap;  and  its  center  o'  will 
be  a point  in  the  valve  circle  whose  center  is  at  o;  this  circle 
can  now  be  drawn,  since  the  radius  o o'  is  known. 

The  diameter  a'b'  is  equal  to  the  required  valve  travel. 
Now,  with  o'  as  a center,  draw’  a circle  tangent  to  o m,  and 
the  radius  of  this  circle  will  be  the  required  inside  lap.  Draw 
0/ through  0'  and  the  angle  fob  is  the  required  angle  of 
advance.  Draw  the  radius  ol  tangent  to  the  inside  lap 
circle  on  its  upper  side,  and  Ip  perpendicular  to  a b. 

Then,  ^ represents  the  fraction  of  the  stroke  at  which 
compression  begins. 

Example  2. — Given,  the  valve  travel,  the  angle  of  advance, 
the  cut-off,  and  the  point  of  compression,  to  find  the  lead 
and  the  outside  and  inside  lap. 

Solution.— Draw  the  crankpin  circle,  as  before,  and  the 
valve  circle  a' eb'\  construct  the  angle/06  equal  to  the  angle 
of  advance.  By  the  same  method  as  employed  in  the  last 
example,  locate  the  radii  od  and  ol,  representing  the  posi- 
tions of  the  crank  at  the  points  of  cut-off  and  compression, 
respectively. 

About  the  point  o',  at  which  0 f intersects  the  valve  circle, 
describe  a circle  tangent  to  od,  and  the  radius  o'j  of  this 
circle  will  be  the  required  outside  lap.  Now  draw  the  line  gh 
parallel  to  a 6 and  tangent  to  the  outside  lap  circle;  then,  the 
perpendicular  distance  between  gh  and  a 6 is  the  required 
lead.  The  radius  of  a circle  drawn  from  o'  tangent  to  o l will 
be  the  inside,  lap. 

Example  3.— Given,  the  valve  travel,  outside  lap,  and  the 
lead,  to  find  the  point  of  cut-off  and  angle  of  advance. 

Solution. — Draw  the  crankpin  circle  and  the  valve  circle 
a'  e b'  as  before;  draw  a line  parallel  to  a 6,  at  a distance  above 
it  equal  to  the  outside  lap  r plus  the  lead,  intersecting  the 
valve  circle  at  the  point  o'.  About  o'  as  center,  and  with  a 
radius  equal  to  the  given  lap,  describe  a circle;  draw  od 


192 


MACHINE  DESIGN. 


tangent  to  this  circle,  and  drop  a perpendicular  from  c2,  meet- 
ing line  ab  at  a point  k;  then  the  required  cut-off  is  represented 
CL  Jc 

by  the  fraction  Draw  the  radius  of  through  the  point  o' 

and  the  angle /o  6 is  the  required  angle  of  advance. 

Example  4.— Given,  the  outside  lap,  the  lead,  and  the  point 
of  cut-off,  to  find  the  valve  travel  and  the  angle  of  advance. 

Solution.— Draw  the  crankpin  circle  as  before,  and  by  the 
same  method  as  employed  in  Example  1 locate  the  radius  o <2, 
the  position  of  the  crank  at  the  point  of  cut-off.  Draw  g h 
parallel  to  a b , and  at  a distance  above  it  equal  to  the  lead. 
At  a distance  above  the  line  ab  equal  to  the  lap  plus  the  lead, 
draw  another  line  parallel  to  ab ; about  a center  o'  on  this 
line,  and  with  a radius  o' j equal  to  the  outside  lap,  describe 
a circle  tangent  to  o d and  g h.  Draw  the  radius  of  through 
o',  then/o  b will  be  the  required  angle  of  advance.  About  o 
as  a center,  and  with  a radius  o o',  describe  the  valve  circle 
af  e b',  and  a'  b'  will  be  the  required  valve  travel. 


LOCKNUTS. 

A good  method  of  locking  a nut  is  shown  in  the  figure. 

The  lower  portion  of  the  nut  is 
turned  down,  and  in  the  center  of 
the  circular  portion  a groove  is  cut. 
A collar  is  fastened  by  means  of  a 
pin  to  one  of  the  pieces  to  be  con- 
nected, and  into  this  collar  is  fitted 
the  circular  part  of  the  nut.  The 
nut  is  then  bound  to  the  collar  by 
a setscrew  passing  through  the 
latter,  the  point  of  the  setscrew  engaging  into  the  groove 
turned  in  the  nut.  The  following  proportions  have  proved 
very  satisfactory,  in  which  <2,  the  diameter  of  the  bolt,  is 
taken  as  the  unit.  All  dimensions  are  in  inches: 
a = l£  <2  — TV';  /=£<*  + £"; 
b = l£<2  + £";  . g = £<2  + ^"; 

c = £<2  + £";  h = £<2  + £". 

e = £c2; 


LINE  SHAFTING. 


193 


PROPORTION  OF  KEYS. 

In  common  designing,  the  sizes  of  keys  are  determined 
by  empirical  formulas,  which  give  an  excess  of  strength.  For 
an  ordinary  sunk  key,  these  proportions  may  be  adopted: 
t — thickness  of  key  in  inches; 
b = breadth  of  key  in  inches; 
d = diameter  of  shaft  in  inches; 
b = id; 
t = *6  = id. 


LINE  SHAFTING. 

The  speed  of  a shaft  is  fixed  largely  by  the  speed  of  the 
driving  belt  or  the  diameters  of  the  pulleys  upon  it.  In 
general,  machine-shop  shafts  run  about  120  to  150  rev.  per 
min.;  shafts  driving  wood-wor king  machinery,  about  200  to 
250  rev.  per  min.;  in  cotton  mills,  the  practice  is  to  make  the 
shaft  diameter  smaller  and  run  at  a higher  speed.  Line  shafts 
should  generally  not  be  less  than  li  in.  in  diameter. 

The  distance  between  the  bearings  should  not  be  great 
enough  to  permit  a deflection  of  more  than  yfoy  in.  per  foot  of 
length;  hence,  the  bearings  must  be  closer  when  the  shaft  is 
heavily  loaded  with  pulleys. 

The  maximum  distances  between  bearings  of  different 
sizes  of  continuous  shafts  used  for  transmitting  power  are: 
Distances  Between  Bearings. 


Diameter  of  Shaft. 
Inches. 

Distance  Between  Bearings  in  Feet. 

Wrought-Iron 

Shaft. 

Steel  Shaft. 

2 

11 

11.50 

3 

13 

13.75 

4 

15 

15.75 

5 

17 

18.25 

6 

19 

20.00 

7 

21 

22.25 

8 

23 

24.00 

9 

25 

26.00 

Pulleys  that  give  out 'a  large  amount  of  power  should  be 
placed  as  near  a hanger  as  possible. 


194 


MACHINE  DESIGN. 


SHAFT  COUPLINGS. 

A box,  or  muff,  coupling  is  shown  in  the  figure.  It  consists 


of  a cast-iron  cylinder 
that  fits  over  the  ends 
of  the  shaft.  The  two 
ends  are  prevented 
from  moving  relatively 
to  each  other  by  the 


sunk  key.  The  key  way  is  cut  half  into  the  box  and  half  into 
the  shaft  ends.  Quite  commonly  the  ends  of  the  shafts  are 
enlarged  to  allow  the  key  way  to  be  cut  without  weakening 
the  shaft. 

The  key  may  be  proportioned  by  the  formula  already 
given.  For  the  other  dimensions,  take 
l = 2£d  + 2" 
t = .4d  + .5" 

Example. — Find  the  dimensions  of  a muff  coupling  for  a 
shaft  2b  in.  in  diameter. 

Solution. — For  the  key  we  use  the  formula  previously 


t = Ad  + .5"  = .4  X 2J  + .5"  = 1 
A flange  coupling  is  shown  in  the  following  figure.  Cast- 


iron  flanges  are  keyed  to  the  ends  of  the  shafts.  To  insure  a 


given, 


For  the  muff, 


PEDESTALS. 


195 


perfect  joint  the  flange  is  usually  faced  in  the  lathe  after 
being  keyed  to  the  shaft.  The  two  flanges  are  then  brought 
face  to  face  and  bolted  together. 

Sometimes  the  ends  of  the  shafts  are  enlarged  to  allow  for 
the  keyway.  To  prevent  the  possibility  of  the  shafts  getting 
out  of  line,  the  end  of  one  may  enter  the  flange  of  the  other. 

The  following  proportions  may  be  used  for  this  form  of 
flange  coupling: 

d = diameter  of  shaft;  n = number  of  bolts. 

D = lfd  +1" 

A = 2£d  + 2" 

l = l*d  + l" 

n=  3 + | 

(Take  the  nearest  whole  number  for  n.) 

A = 1.4  A 
b = %d  + f" 
e = 2b 
t = id 

The  proportions  for  the  key  have  already  been  given. 

In  the  accompanying  figure  is 
shown  a flexible  coupling,  or  uni- 
versal joint.  These  joints,  when 
constructed  of  wrought  iron,  may 
have  the  following  proportions  in 
terms  of  the  diameter  d of  the 
shaft: 

a = 1.8  <2  g = .6  d 

b = 2d  h—  .5  d 

c = d k = .6d 

e = 1.6  d 


PEDESTALS. 

The  names  pedestal , pillow-block , bearing , and  journal-box 
are  used  indiscriminately.  They  are  all  a form  of  bearing, 
and  indicate  a support  for  a rotating  piece. 


196 


MACHINE  DESIGN. 


A form  of  journal-box  frequently  used  for  small  shafts  is 
shown  in  Fig.  1.  It  consists  of  two  parts:  (1)  the  box  that 
supports  the  journal,  and  (2)  the  cap  that  is  screwed  down  to 
the  box.  In  this  journal-box  the  seats  are  of  babbitt,  or,  as  it 
is  commonly  expressed,  the  box  is  babbitted.  The  cap  is  held 
in  place  by  what  are  called  capscrews.  This  is  invariably 
done  in  small  pedestals. 

The  proportioning  of  a pedestal  is  largely  a matter  of 


experience.  Few  or  none  of  the  parts  are  calculated  for 
strength. 

All  the  proportions  of  the  pedestals  that  follow  are  based 
on  the  diameter  of  the  journal  d as  the  unit;  the  length  of 
the  seats  is  the  same  as  that  of  the  journal. 

For  the  journal-box  shown  in  Fig.  1,  the  following  propor- 
tions may  be  used  for  sizes  of  journals  from  £ in.  to  2 in.  diam- 
eter, inclusive.  The  diameter  of  shaft  d is  the  unit. 


PEDESTALS. 


197 


Fig.  2. 


198 


MACHINE  DESIGN. 


m = .25  d + .1875"; 
n = .5  d; 

o = .625"  (constant); 
p = 1.5  d ; 


q = 1.333  <2; 
y = .08  (2; 

s = .125"  (constant); 
* = .16  d; 
u * * 1.333  d; 
v .125(2. 


a = 2.25  (2; 

5 = 1.75  d\ 
c — d; 
c = .375  d; 

/=  .08  d -t-  .0625"; 
g = 1.75  (2; 
h = 2.45  d; 
t = .3  d ; 

J = .33  d; 

= .25  d + .125"; 

Z = .08  d; 

In  Fig.  2 is  shown  a common  form  of  pedestal  that  is  used 
for  somewhat  larger  journals  than  the  one  shown  in  Fig.  1. 

It  consists  of  (1)  a foundation  plate  that  is  bolted  to  the 
foundation  on  which  the  pedestal  rests;  the  plate  is  essential 
when  the  pedestal  rests  on  brickwork  or  masonry,  but  may  be 
dispensed  with  when  the  pedestal  rests  on  the  frame  of  the 
machine;  (2)  the  block  that  carries  the  seats  and  supports  the 
journal;  (3)  the  cap  that  is  screwed  down  over  the  seats.  The 
bolt  holes  in  both  foundation  plate  and  block  are  oblong,  so 
that  the  pedestal  may  be  readily  adjusted. 

The  following  proportions  may  be  used  for  this  kind  of. 
pedestal,  having  journals  from  2 in.  to  6 in.,  inclusive.  An  oil 
cup  having  a £ in.  pipe-tap  shank  may  be  used  on  pedestals 
for  journals  having  diameters  from  3 in.  to  4 in.,  and  f in. 
pipe-tap  shank  for  larger  sizes  up  to  6 in.  diameter. 


Note.— The  shanks  of  oil  cups  and  grease  cups  bought  in 
the  market  are  made  with  a £",  f",  or  £"  pipe  thread. 
The  amount  of  oil  or  grease  the  cup  holds  when  filled  is 
usually  expressed  in  ounces. 


The  diameter  of  journal  d is  the  unit. 


a = 3.25d; 

j = .375(2; 

b = 1.75<2; 

k = 1.0625(2; 

c — c2; 

2 = .875(2; 

e = .5(2; 

m = 1.75(2; 

/ = .4375(2; 

n = 1.25(2; 

g = .09(2; 

o = .125"  (constant); 

h = .3125(2; 

p = .875"  (constant); 

i = .25(2; 

q = .625  <2; 

r = .25  d; 
s = .1875d; 
t = .65(2; 
u = .75  cf; 
v — 1.375(2; 
x = .25(2; 
y = .5<2; 
z = .0625d. 


PEDESTALS. 


199 


Fig.  3. 


200 


MACHINE  DESIGN. 


Fig.  3 shows  a pedestal  suitable  for  the  crank-shaft  of  a 
horizontal  engine  with  journals  from  8 in.  to  20  in.  in  diameter. 
The  block  may  be  complete  in  itself,  as  shown  in  the  figure, 
but  more  often  it  forms  part  of  the  engine  bed. 

The  seats  are  in  three  parts,  and  may  be  adjusted  hori- 
zontally by  means  of  the  wedges  W.  The  lower  seat  may 
be  raised  by  placing  packing  pieces  under  it.  To  obtain  its 
dimensions,  use  the  following  proportions,  which  are  based 
on  the  unit  d = the  diameter  of  the  crank-shaft  journal. 


a = d + 1"; 
b = .5  (2  + 1"; 
c = .66  (2; 
e = .825(2  — .25"; 

/ = .6(2; 
g = .1  d + .5625"; 
h = .1(2  + .25"; 
h'  = .08(2; 
i = .11(2; 

j = .625"  (constant); 
Jc  = .5(2  + 1.25"; 
l = .375"  (constant); 
m = .175(2  + .31.25"; 
n = .25(2  + 25"; 
n'  = .1(2  + .375"; 


q'  = 1.5(2; 
r = .15(2; 
r'  = .1(2; 

ri  — <2; 

s = .9(2; 

2 = 15  (2  + .375"; 

2'  = .9(2; 
u = 1.5(2; 
v = .25(2 + .375"; 
w = 1.45(2; 
u>'  = 1.47(2 
Wi  = 1.75(2; 
x =3  .1(2; 
y = .3  (2 + .75"; 
y'  = .2  (2 + .5"; 
z = .09  (2; 

^ = 2.5"  (constant). 


o = 1"  (constant); 
p = .25(2  + .625"; 

(?  = 1.75(2; 


Taper  of  adjusting  wedge,  1 : 10. 

Further  details  of  the  bottom  seat  and  the  cap  are  shown 
in  Fig.  4,  in  which  the  unit  is  the  same  as  in  Fig.  3,  and  the 
proportions  are  as  follows: 


a = 1"  (constant);  c = .08(2; 

b = 1.65(2— .5";  d = .1(2. 

The  foundation  casting,  or  the  bed  casting,  is  shown  in 
Fig.  5,  and  has  dimensions  to  suit  the  pedestal  that  is  shown 
in  Fig.  3.  The  proportions  of  the  casting  are  given  in  con- 
nection with  Fig.  5,  on  page  201.  The  diameter  d of  the 
crank-shaft  journal  is  taken  as  the  unit. 


PEDESTALS. 


201 


b = 


o'  = 


t = 


2.45  (2  +7.25"; 
2.3  d + 5.25"; 
.5(2  + 3.5"; 
3.5(2  + 2"; 
.25(2  + .5"; 

.25  (2  + 1.75"; 
.25  d + 2.25"; 
.05  d + .5"; 

.05  c?  + 1.125"; 
.05  d + .75"; 
.25  <2  + .75"; 
Ad) 

.6  d) 

1.55  d + 2.5"; 
.25  d + 2"; 
.25(2  + .5"; 

.5  <i  + 4.5"; 
.08(2; 

1.5  (2; 

.15  (2  + .375"; 
.15  (2  + .375"; 
.9(2;  ' 

.15(2  + .875"; 
.2(2; 

1.5(2; 


Fig.  4. 


Fig.  5. 


202 


MACHINE  DESIGN. 


HANGERS. 

A hanger  is  used  when  a shaft  bearing  is  to  he  suspended 
from  the  ceiling.  The  figure  on  page  203  shows  a form  of 
hanger  made  by  a leading  manufacturing  company. 

The  frame  of  the  hanger  is  divided  and  the  parts  are  con- 
nected by  bolts.  With  such  a form,  the  shaft  may  be  more 
easily  removed  than  when  the  hanger  frame  is  a solid  piece. 

The  units  for  determining  the  leading  dimensions  of  a 
shaft  hanger  are  the  diameter  cZ  of  the  shaft  and  the  drop  D 
of  the  hanger. 


The  following  proportions  are 
from  l\  in.  to  4£  in.  in  diameter: 
A = 6 eZ  + .45  D; 

A&  = 2cZ  -J-  .03  D; 

B = 4 d + .35  D; 

C = 2eZ  + .3D; 

E = 2cZ  + .25  D; 

F = .5  cZ  + .01 D; 

Fi  = 1.5  d + .05  D; 

0 = 1.25  d; 

II  = 2d; 

1 = Ad; 

J = .125  d + .01 D; 

K = .5<Z  + .5"- 
L = .2b  d + .5"; 

M = .75  d + .6875"; 

N = .25  d + .375"; 

0 = 1.25  d; 

01  = .094  d + .002  D; 

P = .375  d + .008  D; 

Q = .375  d + .008  D; 

P and  Ri  (see  note) ; 

S = .25  d + -005  D; 

Si  = .125  d + .003  D; 

T = .125  d + .01 D; 

Ti  = (see  note); 

U = 2d; 

V = .bd; 

W ■=  .75  d; 


suitable  for  shafts  ranging 


X 

= 

.375  d; 

Y 

= 

.25cZ  + .125"; 

Z 

= 

.625  d; 

a 

= 

.lbd  + .375"; 

ai 

= 

2.4  d + .3125"; 

6 

.08  d; 

c 

= 

.125  d + .0625"; 

€ 

= 

.2d; 

ei 

= 

Ad; 

e2 

= 

.2d; 

f 

= 

.375  cZ  -f  1"; 

/i 

= 

.09  d + .25"; 

9 

= 

.lbd; 

9\ 

= 

1.3125  d + .125"; 

h 

= 

1.25  dp. 1875"; 

i 

= 

.1  cf; 

3 

= 

.25  cZ  + .25"; 

3\ 

= 

.125  d + .0625"; 

k 

= 

2.2  d; 

l 

= 

4 d; 

m 

= 

1.4  dp. 375"; 

n 

= 

d; 

0 

= 

.25  d; 

Ol 

= 

.0625  d; 

P 

= 

d; 

Pi 

= 

.0625  d; 

Q 

= 

Ad; 

HANGERS. 


203 


qi  = .15  d; 
r = 2.125  d; 
s = 1.5  d; 

Si  = .125  d; 
t = 2d; 
ti  = .5  d; 
t%  = d; 
t3  = .25  d; 
w = .95  d; 

= .85  d; 

v = .25  d + .125"; 
i\  = .5  d; 


w = d: 

Wi  = .125"  (constant); 
x = .25  d; 

% = d; 

iCo  = 4d  + 2"; 

y = 1.25  d; 

2/i  = .75  d + .0625"; 

2/2  = .4  d + .0625"; 
z = .06  d + .75"; 

Z\  = .12  d + .75"; 

Z2  = .3125"  (constant), 


Thread  of  plugs,  .5  in.  pitch  for  all  sizes. 


204 


MACHINE  DESIGN. 


Note.— To  find  Ru  draw  the  arc  J;  also,  draw  the  arc  Q 
tangent  to  P;  then,  draw  a straight  line  tangent  to  these  arcs, 
and  Pi  will  be  the  distance  along  the  center  line  determined 
by  B included  between  this  tangent  and  the  upper  face  of  the 
hanger.  Having  found  Pb  make  P equal  to  it. 

The  radius  T\  is  made  equal  to  three-eighths  of  the  thick- 
ness at  the  middle. 

The  steps  of  the  ball-and-socket  bearings  are  of  cast  iron, 
and  are  bored  to  fit  the  journal  without  using  either  lining  or 
brasses.  The  ball  and  the  recesses  in  the  ends  of  the  plugs, 
into  which  the  ball  is  fitted,  should  be  faced.  The  screw 
threads  on  the  plugs  may  be  cast  on  the  plugs  or  turned,  the 
latter  being  preferable.  It  is  customary  to  use  2 threads  per  inch 
for  all  sizes  of  plugs.  

BELT  PULLEYS. 

The  accompanying  table  gives  the  dimensions  of  a set  of 
cast-iron  belt  pulleys  ranging  from  6 in.  to  72  in.  in  diameter,  as 

made  by  a well-known 
manufacturing  com- 
pany. These  pulleys 
are  so  designed  that  the 
number  of  patterns 
may  be  kept  within 
reasonable  limits,  and 
at  the  same  time  have 
the  dimensions  corre- 
spond as  nearly  as  pos- 
sible with  well-estab- 
lished rules. 

The  letters  over  the 
columns  of  dimensions 
given  in  the  table  cor- 
respond to  the  letters 
in  the  figure. 

In  all  cases  the  num- 
ber of  arms  is  6,  and  the  arms  increase  in  size  toward  the  hub, 
the  taper  being  £ in.  per  ft. 

In  order  to  prevent  heavy  stresses  in  shafts  and  bearings, 
pulleys  that  are  to  run  at  high  speeds  must  be  carefully 


BELT  PULLEYS. 


205 


balanced.  Perfect  balance  involves  two  conditions:  (a)  the 
center  of  gravity  of  the  pulley  must  lie  in  the  center  line  of 
the  shaft,  (6)  the  straight  line  joining  the  centers  of  gravity 
of  any  pair  of  opposite  halves  of  the  pulley  must  be  perpen- 
dicular to  the  center  line  of  the  shaft. 

The  usual  method  of  balancing  a pulley  is  to  rivet  a weight 
to  the  light  side  and  test  the  balance  by  putting  the  pulley 
on  a mandrel  that  is  placed  on  two  carefully  leveled  ways  on 
which  it  can  roll  with  very  little  friction.  If  the  center  of 
gravity  of  the  pulley  lies  in  the  center  of  the  shaft,  the  pulley 
will  stay  in  position  when  stopped  with  any  point  of  its  cir- 
cumference over  the  mandrel;  if,  however,  one  side  of  the- 
pulley  is  heavier,  the  mandrel  will  roll  until  the  heavy  side 
is  at  the  lowest  possible  point. 

While  the  above  method  does  not  determine  whether  or 
not  the  second  condition  of  perfect  balance  is  fulfilled,  it  is 
generally  sufficient  for  pulleys  running  at  ordinary  limits  of 
speed  and  reasonably  well  made. 

In  some  cases,  however,  a failure  to  meet  the  requirements 
of  the  second  condition  of  perfect  balance  may  result  in  un- 
satisfactory running  and  severe  stresses  in  the  shaft  and  its 
bearings.  Consider  a pulley  in  which  the  center  of  gravity 
of  one  half  is  at  the  right  of  a line  perpendicular  to  the  center 
line  of  the  shaft  while  the  center  of  gravity  of  the  opposite 
half  is  on  the  left  of  the  perpendicular.  This  condition  will 
not  affect  the  balance  of  the  pulley  when  tested  by  the  man- 
drel rolling  on  the  ways;  when,  however,  the  pulley  revolves 
around  the  center  line  of  the  shaft,  the  centrifugal  forces  of 
the  two  halves  act  in  opposite  directions  and  along  different 
lines.  These  forces  thus  form  a couple  that  tends  to  bend 
the  shaft.  Since  the  centrifugal  force  is  proportional  to  the 
square  of  the  number  of  revolutions,  it  is  apparent  that,  at 
high  speeds,  the  bending  effect  may  be  considerable,  even 
though  the  lack  of  symmetry  is  not  very  great. 

It  is  usually  considered  unsafe  to  run  a cast-iron  pulley, 
gear-wheel,  or  flywheel  at  a higher  rim  speed  than  100  ft.  per  sec. 
Since  the  centrifugal  force  increases  in  direct  proportion  to 
the  cross-section  of  the  rim,  it  is  evident  that  it  is  useless  to  try 
to  provide  against  it  by  putting  more  material  in  the  rim. 


206 


MACHINE  DESIGN. 


Proportions  of  Pulleys. 


s 

<a 

d 

o 

Rim. 

Arm. 

Hub. 

Boss. 

S 

Ph 

A 

B 

C 

D 

£ 

F 

G 

PT 

1 

6" 

4 

34 

IS 

37 

TS 

3 

% 

34 

1 

34 

6 

|4 

A 

334 

34 

1 

17 

8 

T3 

334 

34 

34 

1 

34 

10 

3/ 

A 

4 

Jl 

1/ 

1 

12 

34 

+3 

4 

17 

34 

1 

y 

8 

4 

34 

T8b 

if 

A 

3 

n 

34 

1 

34 

6 

8 

10 

12 

4 

3B2 

34 

1*' 

TS 

334 

434 

5>4 

34 

1 

! 

p~ 

10 

34 

T36 

1 5 
T6 

T5 

3 ‘ 

>4 

34 

1 

34 

6 

8 

31 

34 

1* 

gf 





10 

Its 

34 

5% 

% 

134 

12 



12 

4 

31 

34 

1 

334' 

"34" 

"34" 

i 

34 

6 

1/4 

4 

8 

5 

"34 

134 

34 

14 

10 

12 

4 

6 

A+ 

*+' 

T3 

34 

1/4 

134 

14 

"ST 

534 

6/| 

334 

434 

I 

I 

i 

134 

/! 

8 

136 

156 

lie 

5 

10 

6 



16 

12 

4 

31+ 

34 

1H 

1% 

13 

s 

fi 

... 

. 

T" 

34 

6 

% 

* % 

134 

% 

8 

T%+ 

TB6 

l*’ 

% 

5 





10 

6 

....... 

12 

16 

7 

"ii" 

ti 

37 

3l 

1% 

if 

34 

5 4 

I 

18 

4 

TS 

TS 

1* 

A 

4 

% 

% 

34 

6 

434 



8 

31 

51 

134' 

"if 

534 

"54" 

10 

6 



12 

734 

% 

% 

134 

16 

34 

34 

234 

134 

8 



20 

9 

20 

4 

1S  + 

TS 

1% 

"34 

4 

% 

"34" 

134 

34 

6 

434 

8 



5 

10 

31 

'"54" 

6 

12 

7 

134' 



16 

A 

T5 

234' 

134" 

8 

% 

"34 

20 

10 

1 





PROPORTIONS  OF  PULLEYS. 


207 


Table— ( Continued). 


a 

cC 

0> 

o 

Rim. 

Arm. 

Hub. 

Boss. 

A 

A 

B 

C 

x> 

E 

F 

G 

H 

J 

22" 

4 

IS 

IK 

K 

4 

K 

K 

iK 

K 

6 

4K 

k 

8 

5 

10 

*+ 

ii 

1% 

If 

6K 

"K 

.. 



12 

iK 

16 

"s% 

i 

20 

+3 

2K 

IK 

11 

IK 

iK 

24 

4 

32 

4* 

1t93 

ll 

4 

K 

K 

iK 

K 

6 

8 

10 

% 

K~ 

IT 

"iK- 

0/2 

7 

"Vs 

iK 

12 

16 

’9K 

i " 

20 

24 

IS 

32 

2% 

1% 

IK 

K 

iK 

26 

4 

6 

32 

hi 

1U 

“k 

f* 

“k 

iK 

K 

8 

6 

Va 

iK 

10 

K 

2*"" 

7 

12 

7K 

16 

10 

IK 

"Va 

'i K 

20 

24 

TS3  + 

32 

211 

iii 

S* 

4^ 

5K 

28 

4 

6 

372+ 

ii" 

I 

"I 

iK 

iK 

K 

8 



7 

10 

7K 



12 

16 

20 

K+ 

K 

2K 

11 

8 

10 

11 

i 



TS3 

I 

32 

IK 

If 

iK 

"I 

K 

iK 

30 

24 

4 

6 

8 

10 

12 

32+ 

f 

"§ 

f 

"k 

K 

i'K 

iK 

K 

13 

2K 

T ' 

8 

T" 

16 

20 

8K 

UK 

iK 

Va 

iK 



32 

24 

4 

1 

iK 

if 

13 

4K 

iK 

K 

6 

8 

5M 

6% 

l 

10 





7^ 



208 


MACHINE  DESIGN. 


Table— ( Continued). 


a 

c3 

6 

o 

Rim. 

Arm. 

Hub. 

Boss. 

s 

A 

B 

C 

D 

E 

F 

0 

H 

7 

12 

TS 

33 

2^ 

1* 

8 

134 

% 

134 

16 

934 



20 

11 

'134' 



24 

13 

34 

4 

34+ 

34 

234 

15 

TS 

434 

61/ 

Vs 

% 

134 

34 

8 

°/2 

634 

i ’ 

10 

12 

16 

TS 

'W 

2/s 

Its 

1 

934 

134 

134' 



20 

12 

"134' 

24 

13 

36 

4 

g 

34+ 

34 

2t3s 

TS 

434 

534 

"Vs 

"'34 

134' 

8 

m 

....... 

10 

734 



12 

TS 

33 

10 i 

134' 

"Vs 

134 

16 

2t9s 

134 

•134 

20 

12 

40 

24 

8 

T6 

M 

‘2*’ 

1 ' 

f 

134 

1 

1 

34 

1 

3l 

12 

434 

% 

134 

16 

M 

34 

2% 

'l34' 

10 

$ 

20 

1134 

i 

i% 

34 

24 

1534 

A 

44 

8 

12 

16 

S3 

TS 

iy* 

134 

6/1 

g 

% 

134 

34 

33 

% 

3 ‘ 

'i* 

10 

"134' 

rj 

20 

12 

1 

134' 

"34" 

24 

3/4 

234 

134 

15 

48 

8 

12 

393  + 

TS 

1 

'134 

34 

16 

% 

TS 

334 

l/s 

10 

134 

1 

"34“ 

20 

12 

54 

24 

12 

16 

TS+ 

1 6 

33 

3" 

'I* 

15 

9 34 
II34 

I 

134 

i ” 

134 

'34" 

20 

H 

M 

334 

134 

134 



24 

15  ’ 

'1% 

2" 



60 

12 

33 

34 

3ys 

Its 

10 

m 

1 

134 

34 

16 

20 

1/5 

34 

m 

134 

1134 

12/4 

i 

134 

"2" 



24 

15 





ROPE  BELTING. 


209 


Table — ( Continued). 


a 

ci 

o 

w 

Rim. 

Arm. 

Hub. 

Boss. 

5 

A 

B 

g 

D 

E 

F 

G 

H 

I 

O) 

q> 

12 

u 

X 

3ft 

1* 

10 

11 4 

1 

y* 

16 

20 

X 

% 

4 X 

lit 

13^ 

1 X 

2 

24] 

15 



72 

12 

16 

% 

t9s 

We 

IB 

10K 

ill 

I 

"ix 

2 

X 

20 

re 

f§ 

2* 

4 

24 

15 

2 



ROPE  BELTING. 

There  is  a growing  tendency  toward  the  substitution  of 
hemp  and  cotton  ropes  for  belting  and  line  shafting  as  a 
means  of  transmitting  power  in  large  factories  and  shops. 
The  advantages  claimed  for  the  rope-driving  system  are: 

1.  Economy;  for  a rope  system  is  cheaper  to  install  than 
either  leather  belting  or  shafting. 

2.  In  the  rope  system  there  is  less  loss  of  power  by  slipping. 

3.  Flexibility;  that  is,  the  ease  with  which  the  power  is 
transmitted  to  any  distance  and  in  any  direction. 

In  this  country,  a single  rope  is  carried  round  the  pulley 
as  many  times  as  is  necessary  to  produce  the  required  power, 
and  the  necessary  tension  is  obtained  by  passing  the  rope 
round  a tension  pulley  weighted  to  give  the  desired  tension. 

The  ropes  used  in  rope  transmission  are  either  of  hemp, 
manila,  or  cotton.  Manila  ropes  are  mostly  used  in  this 
country.  They  are  of  three  strands,  hawser  laid,  and  may  be 
from  | in.  to  2 in.  in  diameter. 

The  weight  of  ordinary  manila  or  cotton  rope  is  about 
.3  D 2 lb.  per  ft.  of  length,  where  D represents  the  diameter  of 
the  rope  in  inches.  Letting  w = the  weight  per  foot  of 
length,  w — .3  Z)2. 

The  breaking  strength  of  the  rope  varies  from  7,000  to 
12,000  lb.  per  sq.  in.  of  cross-section.  The  average  value  may 
be  taken  as  7,000  Z)2.  when  D is  the  diameter  of  rope. 


210 


MACHINE  DESIGN. 


II  - 


For  a continuous  transmission,  it  has  been  determined  by- 
experiment  that  the  best  results  are  obtained  when  the 
tension  in  the  driving  side  of  the  rope  is  about  ^ of  the 
breaking  strength.  That  is, 

7 000  T) 2 

Ti  = tension  in  tight  side  = -h — — = 200  D2. 

The  ropes  run  in  V-shaped  grooves,  and  the  coefficient  of 
friction  is,  of  course,  greater  than  on  a smooth  surface.  The 
coefficient  for  grooves  with  sides  at  an  angle  of  45°  may  be 
taken  at  from  .25  to  .33. 

The  horsepower  that  can  be  transmitted  by  a single  rope 
running  under  favorable  conditions  is  given  by  the  formula 

'=^(200--^-V 

825  V 107.2/ 
in  which  H = horsepower  transmitted; 

D = diameter  of  rope  in  inches; 
v = velocity  of  rope  in  feet  per  second. 

The  maximum  power  is  obtained  at  a speed  of  about  84  ft. 
per  sec.  For  higher  velocities,  the  centrifugal  force  becomes 
so  great  that  the  power  is  decreased,  and  when  the  speed 
reaches  145  ft.  per  sec.  the  centrifugal  force  just  balances  the 
tension,  so  that  no  power  at  all  is  transmitted.  Consequently, 
a rope  should  not  run  faster  than  about  5,000  ft.  per  min.,  and 
it  is  preferable  on  the  score  of  durability  to  limit  the  velocity 
to  3,500  ft.  per  min. 

Example.— A rope  flywheel  is  26  ft.  in  diameter,  and 
makes  55  rev.  per  min.  The  wheel  is  grooved  for  35  turns  of 
1¥'  rope.  What  horsepower  may  be  transmitted  ? 

Solution.— Velocity  in  feet  per  second  = 


26  X 7T  X 55 

v = : 

60 

Applying  the  formula, 

v D2 1 


H = 


2 / V2  \ 

825  I 200"  107.2)- 

the  horsepower  transmitted  by  one  rope  or  turn  is 
74.9  X (H)2  / 90n  _ (74.9)2 \ 

825  \200  107.2 

Then,  30.16  XS5  = 1,055.6  = horsepower  transmitted  by 
the  35  ropes. 


')= 


30.16. 


ROPE  BELTING. 


211 


Example.— How  many  times  should  a 1"  rope  be  wrapped 
around  a grooved  wheel  in  order  to  transmit  200  horsepower, 
the  speed  being  3,500  ft.  per  min.? 

Solution.—  3,500  ft.  per  min.  = = 58|  ft.  per  sec. 


Applying  the  formula,  the  horsepower  transmitted  with 
one  turn  is, 


Hence,  200  -r- 11.9  = 16.8,  say  17  turns. 

Rope  pulleys  differ  from  belt  pulleys  only  in  their  rims. 
The  inclination  of  the  sides  of  the  grooves  may  vary  from  30° 
to  60°.  The  more  acute  the  angle,  the  greater  the  coefficient 
and,  consequently,  the  wear  on  the  rope. 

A section  of  a grooved  rim  in  which  the  sides  of  the 
grooves  are  formed  with  circular  arcs  is  shown  in  the  figure. 


The  proportions  for  this  rim  are  as  follows,  using  the 
diameter  D of  the  rope  as  a unit: 


a = ££>; 
b = ZD  + rV'l 
c = D; 
d = 1.6  D; 


e = i 2)  + TV'; 
/=  i2>  + iV'; 
g = 

h = \D  + 


The  radii  ri  and  r2  are  to  be  found  by  trial;  they  should 
be  of  such  lengths  as  to  make  the  curves  drawn  by  them 


tangent  to  the  required  lines. 


212 


MACHINE  DESIGN. 


The  long  radius  R is  determined  by  drawing  a line  through 
the  center  of  the  rope  at  an  angle  of  22£°  with  the  horizontal, 
and  producing  it  until  it  intersects  a line  drawn  through  the 
tops  of  the  dividing  ribs;  then,  with  this  point  of  intersection 
as  a center,  draw  the  curve  forming  the  side  of  the  groove 
tangent  to  the  circumference  of  the  rope. 

The  advantage  claimed  for  this  groove  is  that  the  rope 
will  turn  more  freely  in  it,  thus  presenting  new  sets  of  fibers 
to  the  sides  of  the  grooves  and  increasing  the  life  of  the  rope. 

The  diameter  of  a rope  pulley  should  be  at  least  30 
times  the  diameter  of  the  rope.  Good  results  are  obtained 
when  the  diameters  of  pulleys  and  idlers  on  the  driving  side 
are  40  times,  and  those  on  the  driven  side  30  times,  the  rope 
diameter.  Idlers  used  simply  to  support  a long  span  may 
have  diameters  as  small  as  18  rope  diameters,  without 
injuring  the  rope. 

When  possible,  the  lower  side  of  the  rope  should  be  the 
driving  side,  for  in  that  case  the  rope  embraces  a greater 
portion  of  the  circumference  of  the  pulley,  and  increases  the 
arc  of  contact. 

When  the  continuous  system  of  rope  transmission  is  used, 
the  tension  pulley  should  act  on  as  large  an  amount  of  rope 
as  possible.  It  is  good  practice  to  use  a tension  pulley  and 
carriage  for  every  1,200  ft.  of  rope,  and  have  at  least  10$  of  the 
rope  subjected  directly  to  the  tension. 

Aside  from  the  grooved  rim,  rope  pulleys  are  constructed 
the  same  as  other  pulleys.  They  may  be  cast  solid,  in  halves, 
or  in  sections.  The  pulley  grooves  must  be  turned  to  exactly 
the  same  diameter;  otherwise,  the  rope  will  be  severely 
strained.  

TRANSMISSION  OF  POWER  BY 
WIRE  ROPE. 

Wire  rope  for  transmitting  power  is  made  up  of  6 strands 
twisted  about  a hemp  core,  each  strand  being  composed  of 
either  7 or  19  wires,  according  to  the  size  of  the  sheaves,  the 
19-wire  rope  being  employed  in  cases  where  it  is  impracticable 
to  use  the  larger  sheaves  required  by  the  7-wire  rope.  Where 
the  conditions,  however,  do  not  preclude  the  use  of  the 


WIRE  ROPE. 


213 


proper  size  of  sheaves,  the  7-wire  rope  is  to  be  recommended 
in  preference  to  the  other,  except  sometimes  on  very- 
short  spans,  where  19-wire  rope  is  to  be  preferred,  composed 
of  the  same  size  of  wires  as  the  smaller  7-wire  rope,  such  as 
would  ordinarily  be  used  to  transmit  the  power,  and  run 
under  a tension  corresponding  to  the  smaller  rope,  or  con- 
siderably below  the  maximum  safe  tension  of  the  rope  used. 
This  is  done  in  order  to  avoid  stretching,  which  would  other- 
wise occur,  and  the  consequent  use  of  mechanical  appliances 
for  preserving  the  necessary  tension. 

In  flying  transmission,  where  the  rope  makes  a single  half 
lap  at  each  end,  the  sheaves  are  usually  made  of  cast  iron, 
with  rims  having  grooves  lined  with  segments  of  rubber  and 
leather,  dipped  in  tar,  and  laid  in  alternately,  upon  which 
the  rope  tracks.  The  diameters  of  the  minimum  sheaves, 
corresponding  to  a maximum  efficiency,  are  as  follows, 
according  to  a prominent  manufacturer: 

Diam.  of  sheave  for  7-wire  steel  rope,  77  times  diam.  of  rope. 
Diam.  of  sheave  for  19-wire  steel  rope,  46  times  diam.  of  rope. 
Diam.  of  sheave  for  7-wire  iron  rope,  160  times  diam.  of  rope. 
Diam.  of  sheave  for  19- wire  iron  rope,  96  times  diam.  of  rope. 

In  long-distance  transmissions,  where  the  rope  makes  2 or 
more  half  laps  at  each  end  about  a pair  of  drums  or  several 
sheaves,  the  rims  may  be  lined  with  wood  or  the  rope  may  be 
run  in  plain  turned  grooves. 

The  horsepower  capable  of  being  transmitted  is  deter- 
mined by  the  general  formula: 

N = [c  D2  — .000006  (w  + g\  + g^)]v , 

in  which 

D — diameter  of  rope  in  inches; 
v = velocity  of  rope  in  feet  per  second; 
w = weight  of  rope  in  pounds; 

<7i  = weight  of  terminal  sheaves  and  shafts; 
g>z  = weight  of  intermediate  sheaves  and  shafts; 
c = constant  depending  on  the  material  of  which 
rope  is  made,  the  character  of  the  filling  or  sur- 
face material  in  the  sheaves  or  drums  upon 
which  the  rope  tracks,  and  the  number  of  half 
laps  at  each  end. 


214 


MACHINE  DESIGN. 


The  values  of  c for  from  1 up  to  6 half  laps  for  steel  rope 
are  given  in  the  following  table: 


c for  Steel 

Number  of  Half  Laps  at  Each  End. 

Rope  on 

1 

2 

3 

4 

5 

6 

Iron. 
Wood. 
Rubber  and 
Leather. 

5.61 

6.70 

9.29 

8.81 

9.93 

11.95 

10.62 

11.51 

12.70 

11.65 

12.26 

12.91 

12.16 

12.66 

12.97 

12.56 

12.83 

13.00 

The  values  of  c for  iron  ropes  are  one-half  the  above.  It 
is  apparent  from  this  table  that,  when  more  than  3 half  laps 
are  made,  the  character  of  filling  or  surface  in  contact  is 
immaterial  so  far  as  slipping  is  concerned. 

Where  the  distance  is  comparatively  short,  as  in  most 
flying  transmissions,  the  effect  of  the  weight  of  the  rope  and 
sheaves  is  so  slight  that  it  may  be  neglected,  and  we  have 
the  general  rule,  that  the  actual  horsepower  capable  of  being 
transmitted  by  a wire  rope  approximately  equals  c times  the 
square  of  the  diameter  of  the  rope  in  inches,  multiplied  by  the 
speed  of  the  rope  in  feet  per  second. 

The  tension  of  the  rope  is  measured  by  the  amount  of  sag 
or  deflection  at  the  center  of  the  span,  and  the  deflection 
corresponding  to  the  maximum  safe  working  tension  is  deter- 
mined by  the  following  formulas,  in  which  s represents  the 
span  in  feet: 

Deflection.  Steel  Rope.  Iron  Rope. 

Still  rope  at  center,  in  ft h = .00004  s2  h = .00008  s2 

Driving  portion,  running,  in  ft...  hi  = .000025s2  hi  = .00005s2 

Slack  portion,  running,  in  ft .../^  = .0000875s2  h ? = .000175s2 

In  very  long  transmissions  it  often  happens  that  the  con- 
ditions will  not  allow  of  the  required  amount  of  tension  to 
drive  properly  with  but  a single  half  lap  on  the  pulley.  In 
such  cases  it  is  customary  to  give  the  rope  a sufficient  num- 
ber of  half  turns  around  successive  grooves  in  the  driving 
pulley  and  a series  of  guide  pulleys  that  serve  to  lead  the 
rope  from  one  groove  on  the  driving  pulley  to  the  next. 

With  this  arrangement  a guide  pulley  at  one  end  of  the 


PIPE  FLANGES. 


215 


line  is  usually  made  to  serve  the  purpose  of  a tension  pulley 
by  being  mounted  in  a movable  frame  that  can  be  drawn  by 
means  of  a screw  or  a weight  so  as  to  give  the  rope  the 
desired  tension.  


PIPE  FLANGES. 

The  figure  shows  the  method  of  flanging  and  bolting  the 
ends  of  two  cast- 
iron  pipes.  The 
dimensions  of  the 
flanges  for  the 
various  sizes  of 
pipes  are  given 
in  the  following 
table: 

Standard  Pipe  Flanges. 


n = number  of  bolts. 


a 

b 

c 

d 

n 

e 

/ 

9 

2.0 

.409 

% 

2.000 

4 

Vs 

4.75 

6.00 

2.5 

.429 

% 

2.250 

4 

ji 

5.50 

7.00 

3.0 

.448 

VS 

2.500 

4 

k 

6.00 

7.50 

3.5 

.466 

vs 

2.500 

4 

Ji 

7.00 

8.50 

4.0 

.486 

k 

2.750 

4 

it 

7.50 

9.00 

4.5 

.498 

ft 

3.000 

8 

if 

7.75 

9.25 

5 

.525 

% 

3.000 

8 

it 

8.50 

10.00 

6 

.563 

3/ 

3.000 

8 

1 

9.50 

11.00 

7 

.600 

74 

3.250 

8 

Its 

10.75 

12.50 

8 

.639 

ft 

3.500 

8 

ix 

11.75 

13.50 

9 

.678 

% 

3.500 

12 

IX 

13.25 

15.00 

10 

.713 

3.625 

12 

lTs 

14.25 

16.00 

12 

.790 

vs 

3.750 

12 

l|f 

17.00 

19.00 

14 

.864 

1 

4.250 

12 

1% 

18.75 

21.00 

15 

.904 

1 

4.250 

16 

1% 

20.00 

22.25 

16 

.946 

1 

4.250 

16 

Its 

21.25 

23.50 

18 

1.020 

Ws 

4.750 

16 

1t9s 

22.75 

25.00 

20 

1.090 

VA 

4.750 

20 

lit 

25.00 

27.50 

22 

1.180 

134 

5.500 

20 

lit 

27.25 

29.50 

24 

1.250 

1X 

5.500 

20 

lfl 

29.50 

32.00 

26 

1.300 

IX 

5.750 

24 

2 

31.75 

34.25 

28 

1.380 

134 

6.000 

28 

2txs 

34.00 

36.50 

30 

1.480 

13Z 

6.250 

28 

2ps 

36.00 

38.75 

36 

1.710 

1/i 

6.500 

32 

2% 

42.75 

45.75 

42 

1.870 

IX 

7.250 

36 

2/4 

49.50 

52.75 

48 

2.170 

i/t 

7.750 

44 

56.00 

59.50 

216 


MACHINE  DESIGN. 


LINING  FOR  SEATS. 

Seats  for  large  bearings  are  often  lined  with  Babbitt  metal, 
or  anti-friction  metal.  It  has  been  found  by  experience  that 
a bearing  will  run  cooler  when  so  lined,  probably  because 
the  Babbitt  metal,  being  softer,  accommodates  itself  to  the 
journal  more  readily  than  the  more  rigid  gun  metal. 

Some  of  the  common  methods  of  lining  the  seats  are 
shown  in  the  figure.  At  (a)  the  Babbitt  metal  is  shown  cast 


into  shallow  helical  grooves;  at  (&),  into  a series  of  round 
holes;  and  at  (c),  into  shallow  rectangular  grooves.  Conse- 
quently, the  journal  rests  partly  on  the  brass  and  partly  on 
the  Babbitt  metal. 

In  cheap  work,  very  frequently  the  seats  are  made  entirely 
of  Babbitt  metal.  A mandrel  the  exact  size  of  the  journal  is 
placed  inside  the  bearing,  and  the  melted  Babbitt  metal  is 
poured  around  it.  In  better  work  a smaller  mandrel  is  used, 
and  the  metal  is  hammered  in,  the  bearing  being  then  bored 
out  to  the  exact  size  of  the  journal. 


CYLINDERS  AND  STEAM  CHESTS. 

Fig.  1 shows  a cylinder  designed  for  a simple  slide-valve 
engine.  The  front  head  A is  cast  solid  with  the  cylinder. 
The  method  of  fastening  to  the  frame  B is  clearly  shown. 

The  principal  dimensions  of  this  cylinder  may  be  deter- 
mined from  the  following  proportions: 

D = diameter  of  cylinder; 

L = length  of  stroke  + thickness  of  piston  + twice  the 
'•piston  clearance; 

C = length  of  stroke  + distance  from  outer  edge  to  outer 
edge  of  piston  rings  — (.01 D + .125"); 
a = 5.5  i; 


CYLINDERS  AND  STEAM  CHESTS. 


217 


b = 4.2  i; 
c = i ; 
d = i; 

e'  = net  area  of  a single  cylinder-head  bolt  whose  nominal 

A P 

diameter  is  e = . , 

4,000 n 

where  A = area  of  cylinder  head  in  square  inches; 
P = steam  pressure; 
n = number  of  bolts. 

The  pitch  of  the  bolts  may  be  from  4.5  to  5.5  in.,  but  should 
never  be  more  than  5 /. 

/ = 1.5  v, 

g = .04  D + .125".  Take  the  nearest  nominal  size  pipe  tap. 
h = twice  the  outside  diameter  of  drain  pipe. 
i ■—  .0003  PD  + .375",  where  P is  the  steam  pressure.  If 
the  steam  pressure  is  less  than  100  lb.,  make  P = 100. 
3 = .85  i; 
k = 4i; 

l = .75  i;  ' 

m = 1.01  D + .125"; 

n = m + 6e,  never  less.  Here,  e is  the  nominal  diameter 
of  the  bolt. 

o = the  nominal  diameter  of  steam-chest  bolts.  The  net 

A'P 

area  of  a single  steam-chest  bolt  = — — r — -» 

& 4,000  n' 

where  A ' = area  of  steam  chest; 

n'  = number  of  bolts*ln  steam  chest. 


p = 2.75  o; 
q = 1.5  r; 
r = 1.25  i; 

s = i.  This  is  required  only  when  the  length  of  the  port 
is  greater  than  12  in. 

t = 1.25  i.  When  D is  greater  than  24  in.,  use  4 bolts  in 
the  standard  and  make  t = 1.1  i. 
u=  1.5  i; 

v — .25"  (constant). 

The  dimensions  of  the  steam  ports , exhaust  ports , and  other 
steam  passages  depend  on  the  velocity  of  the  flow  of  steam. 
The  ports  and  passages  must  be  large  enough  to  allow  the 
steam  to  follow  up  the  advancing  piston  without  loss  of 


218 


MACHINE  DESIGN. 


Fig. 


CYLINDERS  AND  STEAM  CHESTS. 


219 


pressure.  The  maximum  allowable  velocity  of  the  steam  in 
the  passages,  when  they  are  short,  is  about  160  ft.  per  sec. 
But,  with  the  ordinary  ratio  between  the  length  of  connect- 
ing-rod and  length  of  crank,  the  average  velocity  is  about 
five-eighths  of  the  maximum.  Hence,  the  allowable  average 
velocities  are  100  to  125  ft.  per  sec.  for  long  and  short  passages, 
respectively. 

Let  l — length  of  port  in  inches; 

6 = breadth  of  port  in  inches; 

A — area  of  cylinder; 

S = average  piston  speed  in  feet  per  second; 

v — average  velocity  of  steam  in  feet  per  second. 

Then,  area  of  port  X velocity  of  steam  = area  of  piston 
X velocity  of  piston,  or  Ibv  = AS;  whence, 

ib-±* 

v 

For  long  indirect  passages,  take  v = 100;  and  for  short 
direct  passages,  take  v = 125. 

The  constant  100  may  be  used  for  v,  when  designing  plain 
slide-valve  engines  of  the  ordinary  type,  which  cut  off  late  in 
the  stroke,  and  125  may  be  used  for  high-speed  engines  with 
early  cut-off,  and  for  the  Corliss  type. 

The  area  of  the  exhaust  port  or  ports  may  be  from  If  to  2* 
times  the  area  of  a steam  port. 

The  area  of  the  cross-section  of  the  steam  pipe  is  approxi- 
mately equal  to  the  area  of  the  steam  port;  likewise,  the  area  of 
the  exhaust  pipe  should  be  equal  to  that  of  the  exhaust  port. 

The  length  l of  the  port  may  be  .6  D to  .9  D for  slide-valve 
engines,  and  about  .9  D to  D for  the  Corliss  type. 

The  height  w,  Fig.  1,  of  the  valve  seat  must  be  such  that 
the  area  of  the  most  contracted  part  of  the  exhaust  port  is  not 
less  than  75$  of  the  area  of  the  steam  port. 


THE  STEAM  CHEST. 

Fig.  2 shows  a steam  chest  for  the  cylinder  illustrated  in 
Fig.  1.  The  principal  dimensions  are  to  be  determined  by 
the  following  proportions,  which  are  based  on  the  thickness 
i of  the  cylinder  walls,  and  on  the  travel  and  dimensions  of 
the  valve: 


220 


MACHINE  DESIGN. 


a = length  of  valve  + travel  of  valve  -f  twice  the  clear- 
ance between  the  valve  and  the  steam  chest  at  ends 
of  valve  travel; 

b = breadth  of  valve  + twice  the  clearance  between  one 
valve  and  steam  chest; 
c = .75  i; 


d = 2.75  o,  where  o is  the  nominal  diameter  of  the  steam- 
chest  bolts,  as  in  Fig.  1; 

e = .04  j/  A'  H-  .125"  for  all  areas  above  100  sq.  in. 
A'  = area  of  steam  chest,  outside  measurement,  in 
square  inches; 

/ = 1.3  e; 
g = .85  i ; 

h — height  of  valve  + necessary  clearance; 
t = .85  i ; 
j = 2.5  i. 

Note.— When  the  area  of  the  steam-chest  cover  is  less  than 
100  sq.  in.,  its  thickness  e may  be  made  equal  to  i.  If  the  area 
of  the  steam-chest  cover  exceeds  600  sq.  in.,  the  height  of  the 
ribs  should  be  3.5  i,  and  their  number  should  be  increased. 


CYLINDERS  AND  STEAM  CHESTS. 


221 


Fig.  3 shows  a design  for  a steam-chest  cover  when  the 
steam-pipe  flange  is  on  one  side  of  the  steam  chest.  Deter- 
mine the  thickness  e-by  the  same  formula  and  rules  as  for  the 
cover  in  Fig.  2.  The  other  dimensions  are  found  as  follows: 
c = .75  e;  j = 2.6  c; 

/ = 1.3  c;  r = 6 c. 

p should  never  exceed  the  distance  in  inches  given  by  the 


/40  C12 


formula  p 

tion  expressing  the 
thickness  of  the 
cover  in  sixteenths 
of  an  inch,  and  pg 
is  the  gauge  boiler 
pressure  in  pounds 
per  square  inch. 

Example.— Find 
the  maximum 
pitch  of  the  ribs 
for  a cover  £§  in. 
thick,  subjected  to 
a steam  pressure  of 
160  lb.  per  sq.  in. 

Solution. 


where  Ci  is  the  numerator  of  the  frac- 


Substituting  in  the  formula  for  p,  we  have 

140  X J f 

= v-5- = V 


;40  X 152 


= 7.5  in. 


pg  \ 160 

Fig.  4 shows  a Corliss  engine  cylinder  that  may  be 
designed  according  to  the  following  proportions: 

D — diameter  of  cylinder. 


a = 1.21  D + 2c + 1.22"; 
b = 2D  + 1.125"; 
c = .048  D; 

& = .079  D; 
d = .17  D; 

c = .0003  P D + .375",  if 
boiler  pressure  is  above 
100  lb.;  otherwise,  c 
= .03  D + .375"; 

/ = .82  c; 


g = -9e; 

h = b +2(c  + p); 
h'  = h; 
i = 1.8  c; 
j = e; 
k = 1.2c; 

l = 1.7  x + 2"  — 1.2  c,  where 
x = diameter  of  piston 
rod; 

V = .32  D,  about; 


222 


MACHINE  DESIGN. 


m 

— 

.25  D; 

u 

n 

= 

.32  D; 

0 

= 

1.25  e; 

V 

P 

= 

1.3  e; 

Q 

= 

.252); 

w 

Q' 

.32  D; 

r 

= 

1.2  e; 

y 

s 

= 

1.5  c; 

z 

= e;  take  diameter  nearest 
standard  size  bolt; 

= 1.2  e;  take  diameter  nearest 
standard  size  bolt; 

= 1.7  x + 2.25",  where  x =* 
diameter  of  piston  rod; 

= D) 

= 1.5  c. 


-Mi 

T 

ex 

1 

/f 

k\ 

1 

i 

^ )mtu 

pan 

PISTONS. 


223 


A is  to  be  made  according  to  proportions  given  on  page  215. 

Bolts  to  be  made  according  to  the  same  table. 

Note. — The  bolts  for  cylinder  heads  are  to  be  calculated 
from  the  formula  given  for  cylinder-head  bolts  in  connec- 
tion with  Fig.  1. 

In  this  cylinder  the  stuffingbox  £ is  a separate  piece  that 
is  to  be  bolted  to  the  cylinder  head. 


CRANK-SHAFTS. 

For  high-speed,  automatic  short-stroke  engines,  the  follow- 
ing formula  corresponds  with  good  practice: 
d = .44  D + 

where  d is  the  diameter  of  shaft  and  D is  the  diameter  of 
cylinders. 

For  the  Corliss  type,  in  which  the  stroke  is  equal  to  or 
greater  than  twice  the  diameter, 

d = .34  D + 2£", 

when  D is  equal  to  or  greater  than  16  in.  When  D is  less 
than  16  in.,  d = %D. 


PISTONS. 

A form  of  piston  that  is  much  used  is  shown  in  the  follow- 
ing figure.  It  consists  simply  of  a hollow  circular  disk  of 


224 


MACHINE  DESIGN. 


The  packing  rings  $,  s are  made  of  cast  iron,  and  are 
split  and  sprung  into  place.  Their  elasticity  causes  them  to 
press  against  the  cylinder  walls  and  thus  prevent  the  leakage 
of  steam. 

The  following  proportions  will  give  dimensions  suitable 
for  this  piston: 

D = diameter  of  cylinder  in  inches; 
a =*  .2D  + 1.5";  e = .75c; 

6 = diameter  of  piston  rod;  r = .5  c; 

6'  — 26;  p = coreplug; 

c = .18/2  D -.1875";  number  of  ribs  = .08(2)  +34). 


CONNECTING-RODS. 

The  figure  shows  a strap-end  connecting-rod.  The  straps 
Ci  and  c2  are  fastened  to  the  ends  of  the  rods  by  means  of  the 
gibs  % and  a2  and  the  cotters  6i  and  62.  The  cotters  are  held 
in  place  by  the  setscrews  Si  and  s2.  Small  steel  blocks  shown 
between  the  ends  of  the  setscrews  and  the  cotters  are  used  to 
prevent  injury  of  the  cotter  by  the  setscrews. 

The  rod,  cotters,  gibs,  and  straps  may  be  made  of  either 
wrought  iron  or  steel.  The  crankpin  brasses  are  shown 
babbitted  and  wristpin  brasses  without  babbitt.  The  brasses 
are  adjusted  by  means  of  the  cotters,  which  draw  the  straps 
farther  on  to  the  rod  when  they  are  driven  in. 

The  dimensions  for  the  rod  are  given  by  the  following 
proportions: 

For  wristpin  end: 

D = diameter  of  cylinder; 
d = .2 D = diameter  of 
wristpin; 

n = .155  2) + .0625"; 


x = - w2  = a factor  for  use 

4 

in  finding  proportions 
below; 

a = .75  d + .125"; 
a'  = .75  d + .125"; 

6 = j/z5x; 


c = .256; 
e = .125  d; 

f = .26  D + .5"  for  cylinders 
to  26"  in  diameter,  and 
/ = .28  D for  cylinders 
above  26"  in  diameter; 
g = 1.3  n; 

.5  s < 
g-c' 

.32  s 
h ’ 


h = 


CONNECTING-RODS. 


225 


k - — • 
1.8  d’ 

l = .375  b ; 
o = .25  5; 


m = 1.35  d for  wristpins  up 
to  3.5"  in  diameter, 
andm  = 1.48  n for  pins 
above  3.5"  in  diameter; 


226 


MACHINE  DESIGN. 


p = .33  6; 
Q 


= 1.125  d for  wristpins  up 
to  3.5"  in  diameter,  and 
q = 4",  constant,  for  pins 
above  3.5"  in  diameter; 

The  taper  of  the  cotter  is  | in.  per  foot. 

Proportions  for  the  crankpin  end: 

D = diameter  of  cylinder  in 
inches; 

d'  = .28  D = diameter  of 


r = n; 
s = .125  d; 
t = 1.35  d; 
u = .02  D + .25"; 
v = .125  d. 


.32jr' 

h 


crankpin; 

nf  = 1.1  n\  (n  = .155  D + 
.0625"); 

x ’ = ~ n'2  = a factor  used 
4 

below; 
a = .75  d'; 
a'  = .75  d'; 

6 = j/  2.5  x'; 
c'  = .25  6; 
e = .125  d'; 

f = .26  D for  cylinder  diam- 
eters up  to  26",  and 
/ = .28  D for  cylinders 
above  26"  in  diameter; 
g = 1.3  n = same  as  wrist- 
pin  end; 


The  taper  of  the  cotter  is  $ i: 


, = x same  as  wristpin 
1.8  d end; 
l = .375  6; 
m = 1.3  d'\ 
o = .25  6; 
p = .33  6; 

q = same  values  as  for 
wristpin  end; 
r = 1.1  n; 
s = .125  d; 

* = 1.35  d'; 
v = .125'  (constant); 
w = .02  Z>  + .0625"; 

where  D = length  of 
rod,  and 

>8  = stroke,  both  in 
inches. 
l.  per  foot. 


ECCENTRIC  AND  STRAP. 

The  figure  shows  an  eccentric  sheave  and  strap,  both  of  cast 
iron.  The  eccentric  sheave  is  cast  solid,  and  must  be  slipped 
over  end  of  shaft.  The  eccentric  rod  is  held  in  a boss  on 
the  strap  by  a cotter.  For  eccentrics  used  with  valve  stems 
£ in.  in  diameter  or  less,  holes  for  bolts  j are  not  to  be  cored. 
A = boss  for  oil  cup;  B = cross-section  of  rib  r. 


ECCENTRIC  AND  STRAP. 


227 


The  proportions  are  as  follows: 
D = diameter  of  valve  stem; 
d = diameter  of  shaft; 

Qi  = d -f-  2 q -}-  2 ft; 
b = 2D  + .125"; 

V = 2.25  D + . 125"; 
c = 1.5  D; 
e = .75  D; 
e'  = .75  D; 

/ = .7  D; 
g = 1.25  d; 
ft  = D + .125"; 
i = .25  D + . 0625"; 
j = area  of  bolt  at  root  of 
thread  = .38  D2;  use 
the  nearest  standard 


l =3\ 

d-\-2q-\-2h-\~  2 f 

m = g : 

m'=  m; 

= D + .125"; 
n'  = D + .125"; 
o = .75  j; 

P — D; 

q — eccentricity; 
r = D; 
s = 1.25  D; 

* = 2.25  D + 1.25"; 
u = D; 
v = 2.25  D; 
v'  = 1.125  D; 


size  bolt; 
f =j  + - 1875; 


ft  = 4D; 


w = 2.5  D; 
x = 2.25./. 


228 


MACHINE  DESIGN. 


STUFFINGBOXES. 


The  stuffingbox  ol 
the  form  shown  in  the 
figure  is  generally  used 
for  small  work,  such  as 
the  spindles  of  valves, 
etc.  The  outside  of  the 
stuffingbox  is  threaded 
to  receive  a hexagonal 
nut  that  fits  over  the 
gland.  As  the  nut  is 
screwed  down,  the 
gland  is  pressed  down- 
wards and  compresses 
the  packing. 

The  proportions 
used  are: 


d — diameter  of  rod; 
a = 2.5  d + .5"; 
b = 1.5<2  + .125"; 
c = 3d  + .25"; 
e = 3.5(2  + 625"; 


/ = d + .125"; 
g = 2d  + .25"; 
h = 1.5  <2  + .25"; 
i = .25(2  + .0625"; 
k = .5  d. 


This  design  may  be  used  for  rods  up  to  H in.  in  diameter. 
Make  the  number  of  threads  per  inch  the  same  as  for  a 
bolt  whose  diameter  is  equal  to  the  diameter  of  the  rod. 


GEARING. 

The  circular  pitch  of  a gear-wheel  is  the  distance  in  inches 
measured  on  the  pitch  circle  from  the  center  of  one  tooth  to 
the  center  of  the  next  tooth. 

If  the  distance  of  the  teeth  of  a gear  thus  measured  were 
2£  in.,  we  would  say  that  the  circular  pitch  was  2£  in. 

Let  P = circular  pitch; 

D = diameter  of  pitch  circle,  in  inches: 

C = circumference  of  pitch  circle,  in  inches; 

N = number  of  teeth; 

7t  = 3.1416. 


GEARING. 


229 


Then, 


P = 


C ttD  C n D 

noi^t  n=poi-f- 


C = PA  or  7T  D. 


Addendum  = .3  P.  Root  = .4  P. 


The  thickness  of  the  teeth  for  a cut  gear  is  equal  to  .5  P, 
and  for  a cast  gear  .48  P. 

The  diametral  pitch  of  a gear-wheel  is  the  name  given  to 
the  quotient  that  is  obtained  by  dividing  the  number  of  teeth 
in  the  wheel  by  the  diameter  of  the  pitch  circle  in  inches;  or, 
the  diametral  pitch  may  be  defined  as  the  number  of  teeth 
on  the  circumference  of  the  gear-wheel  for  1 in.  diameter  of 
pitch  circle. 

A gear  -with  a pitch  diameter  of  5 in.,  and  having  40  teeth 
is  8 pitch;  one  with  the  same  pitch  diameter  and  having 
70  teeth  is  14  pitch. 

In  the  gear  of  8 pitch  there  are  8 teeth  on  the  circumfer- 
ence for  each  inch  of  the  diameter  of  the  pitch  circle;  and  in 
one  of  14  pitch  there  are  14  teeth  on  the  circumference  for 
each  inch  of  the  diameter  of  the  pitch  circle. 

Let  P = diametral  pitch; 

D = diameter  of  pitch  circle,  in  inches; 

N = number  of  teeth; 
d = outside  diameter; 
l = length  of  tooth; 
t = thickness  of  tboth; 


i>.=  * 
D 


N 

jr  n=pd. 


N + 2 2.157 

a ~ P • P ‘ 


_ i-57 

_P* 


The  circular  pitch  corresponding  to  any  diametral  pitch 
may  be  found  by  dividing  3.1416  by  the  diametral  pitch;  and 
the  diametral  pitch  corresponding  to  any  circular  pitch  may 
be  found  by  dividing  3.1416  by  the  circular  pitch. 

(а)  If  the  diametral  pitch  of  a gear  is  6,  what  is  the  cor- 
responding circular  pitch? 

(б)  If  the  circular  pitch  is  1.5708  in.,  what  is  the  corre- 
sponding diametral  pitch? 


, . 3.1416  coo^ . ...  3.1416 

<°>  "IT  “ -5236m-  (6)  1T5708  - 2' 


230 


ELECTRICITY. 


Diametral  Pitches  With  Their  Corresponding  Circular 
Pitches. 


Diametral 
Pitch,  or  Teeth, 
per  Inch  in 
Diameter. 

Correspond- 
ing Circular 
Pitch. 

Diametral 
Pitch,  or  Teeth, 
per  Inch  in 
Diameter. 

Correspond- 
ing Circular 
Pitch. 

1 

3.1416 

8 

.3927 

2 

1.5708 

9 

.3491 

3 

1.0472 

10 

.3142 

4 

.7854 

12 

.2618 

5 

.6283 

14 

.2244 

6 

.5236 

16 

.1963 

7 

.4488 

20 

.1571 

ELECTRICITY, 


PRACTICAL  UNITS. 

The  volt  is  the  practical  unit  of  electromotive  force  or  elec- 
trical pressure.  It  is  that  electromotive  force  which  will  main- 
tain a current  of  1 ampere  in  a circuit  whose  resistance  is 
1 ohm. 

The  electromotive  force  of  a Daniell’s  cell  is  1.072  volts. 

The  ampere  is  the  practical  unit  denoting  the  strength  of  an 
electric  current,  or  the  rate  of  flow  of  electricity.  It  is  that 
strength  of  current  or  rate  of  flow  which  would  be  maintained 
in  a circuit  whose  resistance  is  1 ohm  by  an  electromotive 
force  of  1 volt. 

One  ampere  decomposes  00009342  gram  of  water  (H20)  per 
second;  or  deposits  .001118  gram  of  silver  per  second. 

The  ohm  is  the  practical  unit  of  resistance.  It  is  that  resist- 
ance which  will  limit  the  flow  of  an  electric  current  under  an 
electromotive  force  of  1 volt  to  1 ampere. 

The  legal  ohm  is  the  resistance  of  a column  of  mercury  106 
centimeters  long  and  1 square  millimeter  sectional  area  at  0°  C. 

One  mile  of  pure  copper  wire  in.  in  diameter  has  a 
resistance  of  13.59  ohms  at  a temperature  of  59.9°  F, 


PRACTICAL  UNITS. 


231 


To  makeHhe  significance  of  these  units  clearer,  take  the 
analogous  case  of  water  flowing  through  a pipe  under  a pres- 
sure of  a column  of  water.  The  force  that  causes  the  water  to 
flow  is  due  to  the  pressure  or  head;  the  flow  or  current  of 
water  is  measured  in  gallons  per  minute;  and  the  resistance 
that  opposes  or  resists  the  flow  of  water  is  caused  by  the  fric- 
tion of  the  water  against  the  inside  of  the  pipe. 

In  electrotechnics,  the  electromotive  force  or  electrical 
potential  expressed  in  volts  corresponds  to  the  pressure  or 
head  of  water;  and  the  resistance  in  ohms  to  the  friction  in 
the  pipe. 

The  unit  that  expresses  the  rate  of  transmission  of  electricity 
per  second  is  called  the  ampere , while  the  flow  of  water  is  ex- 
pressed in  gallons  per  minute. 

In  either  case  the  strength  of  current  or  rate  of  flow  depends 
on  the  ratio  between  the  pressure  and  the  resistance;  for,  as 
the  pressure  increases,  the  current  increases  proportionately; 
and  as  the  resistance  increases,  the  current  diminishes. 

This  relation,  as  applied  to  electricity,  was  discovered  by 
Dr.  G.  S.  Ohm,  and  has  since  been  called  Ohm's  law. 

Ohm’s  Law. — The  strength  of  the  current  in  any  circuit  is 
directly  proportional  to  the  electromotive  force  in  that  circuit 
and  inversely  proportional  to  the  resistance  of  that  circuit , i.  e.t 
is  equal  to  the  quotient  arising  from  dividing  the  electromotive 
force  by  the  resistance. 

Let  E = electromotive  force  in  volts; 

R = resistance  in  ohms; 

C = strength  of  current  in  amperes. 

Then  C = f . R = E — CR. 

1C  (J 

Example.— The  electromotive  force  of  a circuit  is  110  volts, 
and  its  resistance  is  55  ohms;  what  is  the  strength  of  current? 

Solution.—  E = 110  volts.  R = 55  ohms.  C = ^ ~ 

xl  55 

= 2 amperes. 

The  unit  by  which  electrical  power  is  expressed  is  called 
the  watt.  It  is  that  rate  of  doing  work  when  a current  of 
1 ampere  is  passing  through  a conductor  under  an  electro* 
motive  force  of  1 volt,  and  is  equal  to  ^ of  a horsepower. 


232 


ELECTRICITY. 


Let 


E = electromotive  force  in  volts; 

C = strength  of  current  in  amperes; 
E = resistance  in  ohms; 

W = power  in  watts; 

H.  P.  = horsepower. 


E 2 

W = EX  C=  C2X  E = -5-. 

K 


H.  P.  = 


EX  0 C2X  E 


E 2 


W 


746  ~ 746  ~ EX  746  ~ 746' 

One  kilowatt  is  equal  to  1,000  watts:  sometimes  abbrevi- 
ated to  K.  W. 


Watt  hour  is  a unit  of  work.  It  is  used  to  indicate  the 
expenditure  of  an  electrical  power  of  1 watt  for  1 hour. 

Example. — The  resistance  of  a lighting  circuit  is  5 ohms 
and  the  electromotive  force  is  110  volts,  (a)  What  is  the 
amount  of  electrical  power  in  watts  required  for  this  current? 
(6)  What  is  the  equivalent  horsepower? 

Solution.—  E = 110.  E — 5. 


E2 
~E 
E 2 

E X 746 


1102 


= 2,420  watts. 


5X  746 

Conductivity  is  the  name  given  to  the  reciprocal  of  the 
resistance  of  any  conductor.  There  is  no  unit  by  which  to 
express  conductivity. 

Note. — The  reciprocal  of  any  number  is  unity  divided  by 
that  number.  Thus,  the  reciprocal  of  2 is  £ or  .5. 


CURRENTS. 


RULES  FOR  DIRECTION  OF  CURRENT,  ETC. 

To  determine  the  direction  of  a current  in  a conductor 
by  the  aid  of  a compass: 

Rule. — If  the  current  flows  from  the  south  pole  over  the  needle 
to  the  north , the  north  end  of  the  needle  will  point  towards  the 
west , as  in  Fig.  1.  If  the  compass  is  placed  over  the  conductor  so 
that  the  current  will  flow  from  the  south  under  the  needle  to  the 
north , the  north  end  of  the  needle  will  point  towards  the  east , as 
in  Fig.  2. 


CURRENTS. 


233 


To  determine  the  polarity  of  an  electromagnet: 

Rule.— In  looking  at  the  face  of  a pole  (Fig.  3),  if  the  current 


Fig.  3. 


^rRBCTTOS  or 

UmSB  OF  FO&CLE. 


flows  in  the  direction  a , of  the  hands  of  a watch , it  will  he  a south 
pole,  and  if  in  the  opposite  direction  h,  it  will  he  a north  pole. 

To  determine  the  direction  of  an^induced  current  in  a 
conductor  that  is  moving  in  a magnetic  field: 

Rule.—  Place  thumb , 
forefinger , and  middle 
finger  of  right  hand,  each 
at  a right  angle  to  the  other 
two,  as  shown  in  Fig.  h; 
if  the  forefinger  shows  di- 
rection of  lines  of  force 
and  the  thumb  the  direc- 
tion of  motion  of  conduc- 
tor, then  the  middle  finger 
will  show  the  direction  of  the  induced  current. 


Fig.  4. 


Note.— The  above  rule 
will  give  the  polarity  of 
a dynamo. 

To  determine  the  di- 
rection of  motion  of  a 
conductor  carrying  a cur- 
rent when  placed  in  a 
magnetic  field: 


234 


ELECTRICITY. 


Rule. — Place  thumb , forefinger , and  middle  finger  of  the  left 
hand , each  at  a right  angle  to  the  other  two , as  shown  in  Fig.  5; 
if  the  forefinger  shows  the  direction  of  the  lines  of  force  and  the 
middle  finger  shows  the  direction  of  the  current , then  the  thumb 
will  show  the  direction  of  motion  of  the  conductor . 

Note.— The  above  rule  will  give  the  polarity  of  a motor. 


DERIVED  OR  SHUNT  CIRCUITS. 

A circuit  divided  into  two  or  more  branches,  each  branch 
transmitting  part  of  the  current,  is  said  to  be  a derived  circuit; 
the  individual  branches  are  in  multiple-arc,  or  parallel  with 
each  other. 

To  find  the  joint  resistance  of  a derived  circuit: 

Rule. — As  the  conductivity  of  any  conductor  is  equal  to  the 
reciprocal  of  its  resistance , then  the  joint  conductivity  of  two  or 
more  circuits  in  parallel  is  equal  to  the  sum  of  the  reciprocals  of 
their  separate  resistances.  The  joint  resistance  of  two  or  more 
circuits  in  parallel  is  equal  to  the  reciprocal  of  their  joint 
conductivity. 

In  a derived  circuit  of  three  branches,  let  r\,  r2,  and  r3  be 
the  resistances  of  the  three  branches,  respectively.  Their 
joint  conductivity,  or  the  sum  of  the  reciprocals  of  their 
resistances,  is 

1,1  ,1  or  r2  r3  -f  rx  r3  + rx  r2 
r\  r % r3  n r2  r3 

Their  joint  resistance  is,  therefore, 

L__ or  rir2r3 

r2  r3  + n r3  + rx  r2*  r2  r3  + rx  r3  + n r2* 
n r2  r3 

The  joint  resistance  of  a derived  circuit  with  but  two 
branches  in  parallel  may  be  thus  expressed: 
product  of  their  resistances 
sum  of  their  resistances 

Example.— The  resistances  of  two  branches  of  a derived 
circuit  are  20  and  30  ohms,  respectively.  Find  their  joint 
resistance. 

Solution. — 

product  of  their  resistances  _ 600  _ 12  o^mg 
sum  of  their  resistances  ~ 50 


WIRING, 


235 


To  find  the  strength  of  current  in  the  separate  branches  of 
a derived  circuit: 

Rule. — A current  is  divided  among  the  branches  of  a derived 
circuit  in  'proportion  to  their  conductivities — i.  e.,  to  the  reciprocal 
of  their  resistances. 

Example.— If  the  resistances  of  the  two  branches  A and  B 
of  a derived  circuit  are  20  and  30  ohms,  respectively,  and  the 
total  current  in  the  main  circuit  is  60  amperes,  what  is  the 
current  in  each  ? The  conductivity  of  A is  ^ and  of  B ^ . 

Solution.— If  Ci  represents  the  current  in  A,  and  C2 
represents  the  current  in  B, 
then,  Ci : C2  = ■ aV 

Hence  £ _ * _ » _ « 

C6’  C2  “ *’°rC2  “ 20  “ 2- 

Now.  Ci  + C2  = 60,  or  C2  = 60  - Ci. 

Ci  _ 3. 

60— Ci  2’ 

Ci  = 36,  and  C2  = 24. 


Substituting, 


WIRING. 


INTERIOR  WIRING. 

A mil  is  a unit  of  length  used  in  measuring  the  diameters 
of  wires,  and  is  equal  to  .001  in. 

A circular  mil  is  a unit  of  area  used  in  measuring  the 

cross-sections  of  wires,  and  is  equal  to  S(l- in- 

The  sectional  area  of  a wire  expressed  in  circular  mils  is 
equal  to  the  square  of  its  diameter  in  mils. 

Let  c.  m.  = circular  mils; 

C = total  current  in  amperes; 
c = current  in  amperes  to  each  lamp; 
n = number  of  lamps  in  multiple; 
v — volts  lost  in  line; 
r = resistance  per  foot  of  wire; 
d = distance  from  dynamo  to  lamps. 

The  resistance  of  1 ft.  of  commercial  copper  wire,  1 mil 
in  diameter,  at  a temperature  of  75°  F.,  is  10.8  ohms. 


236 


ELECTRICITY. 


A 16  c.  p.  (candlepower)  110- volt  lamp  takes  about 
.5  ampere;  a 16  c.  p.  55-volt  lamp  takes  about  1 ampere. 

All  calculations  for  size  of  wire  must  be  checked  by  com- 
paring with  a table  of  safe  carrying  capacity  (see  table  on 
pages  238  and  239),  and  the  current  value  there  given  must 
not  be  exceeded. 

To  find  the  size  of  wire  for  110-volt  circuit  with  16  c.  p. 
lamps: 

v 

r = — v. 
nd 

„ . 10.8  nd 

For  large  cables,  c.  m.  = • — - — . 

Example. — Find  the  size  of  wire  necessary  for  a circuit 
supplying  current  to  50  110-volt  16  c.  p.  lamps,  300  ft.  from 
the  dynamo,  allowing  a loss  of  5 J in  line. 

Solution.— Volts  at  dynamo  = i??  = 115.8. 

.9o 

Volts  lost  in  line  = 115.8  — 110  = 5.8  = v. 

Then’  r = JT3  = 5oHoO  = -000386  0hm  PCT  ft" 


= .386  ohm  per  1,000  ft. 

The  nearest  size  of  wire,  as  given  in  the  table  on  page  238, 
is  No.  6 B.  & S.,  and  its  current  capacity  is  35  amperes;  there- 
fore it  is  safe. 

To  find  the  size  of  wire  for  a 55-volt  circuit  with  16  c.  p. 
lamps: 


For  large  cables,  c.  m.  = 


2 nd' 
21.6  nd 


Example.— What  size  of  wire  should  be  used  for  supplying 
current  to  75  16  c.  p.  lamps  on  a 55-volt  circuit,  the  distance 
from  dynamo  being  230  ft.,  and  line  loss,  4 volts? 

Solution.— 

r = 2 h = 2 X 75  X 230  “ -000116  °hm  P6r ft" 

= .116  ohm  per  1,000  ft. 

By  referring  to  the  table,  (page  238)  the  nearest  wire  is 
found  to  be  No.  IB.  & S.,  and  its  carrying  capacity  is  greater 
than  the  current  (75  amperes)  that  it  is  to  conduct. 


INTERIOR  WIRING. 


237 


To  find  the  size  of  wire  for  any  circuit  on  a 2-wire  system: 
In  general,  r = c^Td’ 

or,  c.  m.  = 10'8  x 2 ^ X C 

V 

Example.— What  wire  should  he  used  to  carry  450  amperes 
a distance  of  600  ft.,  the  allowable  drop  being  60,  and  the 
E.  M.  F.  at  the  end  of  the  circuit  115  volts? 

115 

Solution.— Volts  at  dynamo  = — = 122.3. 

Volts  lost  in  line  = 7.3. 


Then, 


c.  m.  = 


10.8  X 2 X 600  X 450 
7.3 


798,900. 


Comparing  this  number  with  the  table  on  page  239,  giving 
current  capacity  of  cables,  it  will  be  seen  that  it  is  within  the 
prescribed  limits. 

These  formulas  may  be  used  for  feeders,  mains,  branch 
mains,  service  mains,  and  inside  wiring  on  continuous-current 
circuits,  and  for  secondary  wiring  on  alternating  systems. 

To  find  the  size  of  wire  for  a 110-volt  circuit,  3-wire  system, 
16  c.  p.  lamps: 

4 v 

r — — = for  each  wire. 
n d 


For  large  cables, 

2.7  n d . , 

c.  m.  = for  each  wire. 

v 

In  checking  for  carrying  capacity,  remember  that  the  wire 
carries  only  one-half  the  current  that  would  be  used  on  a 
2-wire  system,  as  the  voltage  between  the  outside  conductors 
is  double  the  voltage  at  the  terminal  of  1 lamp. 

Example.— What  should  be  the  size  of  the  conductors  for 
a 3-wire  system,  when  132  110-volt,  16  c.  p.  lamps  are  installed 
at  a distance  of  210  ft.  from  the  source  of  supply,  the  loss 
being  4 volts? 

Solution.— 

r = isl$lio=  •000B77ohmperft-’ 

= .577  ohm  per  1,000  ft. 

This  would  call  for  a wire  between  Nos.  7 and  8.  The 


238 


ELECTRICITY. 


132  X 5 

current  will  be  = 33  amperes;  but  this  is  too  much 

for  the  wire  to  carry,  and  No.  6 B.  & S.  wire  should  be  used, 
notwithstanding  the  somewhat  less  drop  in  volts  that  will 
result. 

For  continuous-current  circuits,  5$  loss  is  usually  allowed, 
with  full  current  - from  the  dynamo  to  the  lamps.  For  long 
distances  a larger  line  loss  may  be  allowed,  if  the  dynamo  is 
wound  for  that  loss. 

Dimensions,  Weight,  and  Resistance  of  Copper  Wire. 


w 

Diameter  in 
Mils  (d). 
lmil  = .001  in. 

Area. 

Weight  and 
Length. 

Resistance  at 
75°  F.  Ohms  per 
1,000  ft. 

Current. 

Amperes. 

B.  & S.  Gauge. 

Circular 

Mils 

(d2). 

Lb. 

per 

1,000 

Ft. 

Ft. 

per 

Lb. 

Exposed.  1 

Concealed. 

0000 

460.000 

211,600.0 

639.33 

1.56 

.049 

300 

175 

0000 

000 

409.640 

167,805.0 

507.01 

1.97 

.062 

245 

145 

000 

00 

364.800 

133,079.0 

402.09 

2.49 

.078 

215 

120 

00 

0 

324.950 

105,592.0 

319.04 

3.13 

.098 

190 

100 

0 

1 

289.300 

83,694.0 

252.88 

3.95 

.124 

160 

95 

1 

2 

257.630 

66,373.0 

200.54 

4.99 

.156 

135 

70 

2 

3 

229.420 

52,634.0 

159.03 

6.29 

.197 

115 

60 

3 

4 

204.310 

41,742.0 

126.12 

7.93 

.248 

100 

50 

4 

5 

181.940 

33,102.0 

100.01 

10.00 

.313 

90 

45 

5 

6 

162.020 

26,250.0 

79.32 

12.61 

.395 

80 

35 

6 

7 

144.280 

20,817.0 

62.90 

15.90 

.498 

67 

30 

7 

8 

128.490 

16,509.0 

49.88 

20.05 

.628 

60 

25 

8 

9 

114.430 

13,094.0 

39.56 

25.28 

.792 

9 

10 

101.890 

10,381.0 

31.37 

31.88 

.999 

40 

20 

10 

11 

90.742 

8,234.1 

24.88 

40.20 

1.260 

11 

12 

80.808 

6,529.9 

19.73 

50.69 

1.589 

30 

15 

12 

13 

71.961 

5,178.4 

15.65 

63.91 

2.003 

13 

14 

64.084 

4,106.8 

12.41 

80.59 

2.526 

22 

10 

14 

15 

57.068 

3,256.7 

9.83 

101.65 

3.186 

15 

16 

50.820 

2,582.9 

7.80 

128.17 

4.017 

15 

5 

16 

17 

45.257 

2,048.2 

6.19 

161.59 

5.066 

17 

18 

40.303 

1,624.3 

4.91 

203.76 

6.388 

10 

18 

19 

35.890 

1,288.1 

3.89 

257.42 

8.055 

19 

20 

31.961 

1,021.5 

3.08 

324.12 

10.158 

5 

20 

INTERIOR  WIRING. 


239 


Carrying  Capacity  of  Cables. 


Area. 

Circular 

Mils. 

Current. 

Amperes. 

Area. 

Circular 

Mils. 

Current. 

Amperes. 

Exposed. 

Concealed. 

Exposed. 

Concealed. 

200,000 

299 

200 

1,200.000 

1,147 

715 

300.000 

405 

272 

1,300,000 

1,217 

756 

400.000 

503 

336 

1,400,000 

1,287 

796 

500.000 

595 

393 

1,500,000 

1,356 

835 

600.000 

682 

445 

1,600.000 

1,423 

873 

700.000 

765 

494 

1,700,000 

1,489 

910 

800.000 

846 

541 

1,800,000 

1,554 

946 

900.000 

924 

586 

1,900.000 

1,618 

981 

1.000.000 

1,000 

630 

2,000,000 

1,681 

1,015 

1,100,000 

1,075 

673 

To  find  the  size  of  wire  on  primary  circuits  for  alternating 
svstem: 

10.8  X 2 d X Cl  v 

~ v ’ ! ~ C1  X 2 <T 

Cl  — the  total  current  in  amperes  on  primary  circuit,  and 
may  be  determined  by  dividing  the  total  current  on  the 
secondary  circuit  by  the  product  of  the  ratio  and  efficiency 
of  conversion. 

The  ratio  is  generally  20  to  1 on  a 1,000- volt  apparatus  when 
using  52-volt  lamps,  and  10  to  1 when  using  100-  to  110-volt 
lamps. 

The  efficiency  of  conversion  can  be  taken  as  95$  in  ordinary 
transformers. 

Example.— If  the  loss  is  5$,  find  the  size  of  wire  necessary 
on  a 1,000- volt  primary  circuit  when  the  distance  between  the 
dynamo  and  transformer  is  2,000  ft.,  and  the  dynamo  is  supply- 
ing cuirent  for  500  16  c.  p.  52-volt  lamps. 

Solution.— 

Volts  at  dynamo  = = 1,052,  nearly. 

Volts  lost  in  line  = 52. 


240 


ELECTRICITY. 


Assume  the  lamp  efficiency  to  be  3.6  watts  per  c.  p.  Then, 
since  the  product  of  amperes  and  volts  gives  watts, 

Current  to  each  lamp  = ^ = 1.11  amperes. 

Current  on  secondary  = 1.11  x 500  = 555  amperes. 


555 

Total  current  on  primary  is  - ■ = 29.21  amperes. 

.yo  x 


Therefore, 


10.8  X 2 d X Cl  10.8  X 4,000  X 29.21 
v ~ 52 


= 24,267. 


And r - C^xYd  - 29^TxT00O  = -000445  ohm  per  ft" or 

.445  ohm  per  1,000  ft.  This  gives  No.  6 B.  & S.  See  page  238. 

For  alternating  systems  under  ordinary  conditions,  5$  loss 
at  full  load  from  dynamo  to  transformer  on  primary  circuit  is 
a maximum,  although  some  dynamos  are  specially  wound 
for  10$  loss.  A loss  of  from  1 $ to  2$  may  be  allowed  on, 
secondary  circuits  from  transformer  to  lamps. 


INCANDESCENT  LAMPS. 


Let  c = current  in  amperes  to  each  lamp; 

E = electromotive  force  in  volts; 

E 

E — — = resistance  of  lamp  when  hot; 
c.  p.  = candlepower  of  lamp; 

W.  per  c.  p.  = watts  per  c.  p.  (often  called  lamp  efficiency). 


™ CXE 

W.  per  c.  p.  = — — . 


746 

The  number  of  candles  per  electrical  H.  P.  = ^ . 

W.  per  c.  p. 


W.  per  c.  p.  X c.  p. 
“ E 


As  the  commercial  efficiency  of  good  dynamos  is  about 
90$,  the  calculations  of  candles  per  electrical  H.  P.  must 
be  multiplied  by  .90  to  give  the  number  of  candles  per 
mechanical  H.  P. 


Lamp  Efficiencies. 


3.1  watts  per  c.  p.,  or  12  lamps,  16  c.  p.,  to  1 mechanical  H.  P. 

3.6  watts  per  c.  p.,  or  10  lamps,  16  c.  p.,  to  1 mechanical  H.  P. 

4.0  watts  per  c.  p.,  or  8 lamps,  16  c.  p.,  to  1 mechanical  H.  P. 


BELL  WIRING. 


241 


No;te.— Lamps  of  an  efficiency  of  3.1  watts  per  c.  p.  should 
not  be  used  where  the  voltage  averages,  for  any  length  of 
time,  more  than  2 f high;  lamps  of  3.6  watts  per  c.  p.  should 
not  be  used  where  the  voltage  averages  more  than  4 j high; 
and  lamps  of  an  efficiency  of  4 watts  per  c.  p.  should  be  used 
where  the  regulation  of  the  plant  receives  little  or  no  atten- 
tion. If  these  cautions  are  not  followed,  the  life  of  the  lamp 
will  be  greatly  diminished. 

Size  of  Wire  for  Arc-Light  Circuits.— For  ordinary  distances, 
or  small  currents,  use  No.  8 B.  & S.  wire.  For  longer  distances, 
or  large  currents,  use  No.  6 B.  & S.  wire. 


BELL  WIRING. 

The  simple  bell  circuit  is  shown  in  Fig.  1,  where  p is  the 
push  button,  b the  bell,  and  c,  c the  cells  of  the  battery  con- 
nected up  in  series. 

When  two  or  more 
bells  are  to  be  rung 
from  one  push  but- 
ton, they  may  be 
joined  up  in  parallel 
across  the  battery 
wires  as  in  Fig.  2 at  a 
and  b,  or  they  may  be  arranged  in  series  as  in  Fig.  3.  The 
battery  B is  indicated  in  each  diagram  by  short  parallel  lines, 


Fig.  2„ 

this  being  the  conventional  method.  In  the  parallel  arrange- 
ment of  the  bells,  they  are  independent  of  each  other,  and  the 
failure  of  one  to  ring  would  not  affect  the  others;  but  in  the 


242 


ELECTRICITY. 


[¥ 


HIH 


Fig.  3. 


series  grouping  all  but  one  bell  must  be  changed  to  a single- 
stroke action,  so  that  each  impulse  of  current  will  produce 
only  one  movement  of  the  hammer.  The  current  is  then 

interrupted  by  the 
vibrator  in  the  re- 
maining bell,  the 
result  being  that  each 
bell  will  ring  with 
full  power.  The  only 
change  necessary  to 
produce  this  effect  is  to  cut  out  the  circuit-breaker  on  all  but 
one  bell  by  connecting  the  ends  of  the  magnet  wires  directly 
to  the  bell  terminals. 

When  it  is  desired  to  ring 
a bell  from  one  of  two  places 
some  distance  apart,  the 
wires  may  be  run  as  shown 
in  Fig.  4.  The  pushes  p,  p' 
are  located  at  the  required 
points,  and  the  battery  and 
bell  are  put  in  series  with 
each  other  across  the  wires 
joining  the  pushes. 

A single  wire  may  be  used 
to  ring  signal  bells  at  each 
end  of  a line,  the  connections 


0 


H'r 


Fig.  4. 


being  given  in  Fig.  5.  Two  batteries  are  required,  B and  B'y 
and  a key  and  bell  at  ‘each  station.  The  keys  k , k'  are  of  the 
double-contact  type,  making  connections  normally  between 


bell  b or  b'  and  line  wire  L.  When  one  key,  as  k , is  depressed, 
a current  from  B flows  along  the  wire  through  the  upper  con- 
tact of  k'  to  bell  b'  and  back  through  ground  plates  G't  G. 


ANNUNCIATOR  SYSTEM. 


243 


rag: 


When  a bell  is  intended  for  use  with  burglar-alarm  appa- 
ratus, a constant-ringing  attachment  may  be  introduced, 
which  closes  the  bell  circuit  through  an  extra  wire  as  soon  as 
the  trip  at  door  or 
window  is  disturbed. 

In  the  diagram,  Fig.  6, 
the  main  circuit, 
when  the  push  p is 
depressed,  is  through 
the  automatic  drop  d 
by  way  of  the  termi- 
nals a,  b to  the  bell 
and  battery.  This 
current  releases  a pivoted  arm  which,  on  falling,  completes 
the  circuit  between  b and  c,  establishing  a new  path  for  the 
current  by  way  of  e,  independent  of  the  push  p. 

For  operating  electric  bells,  any  good  type  of  open-circuit 
battery  may  be  used.  The  Leclanch6  cell  is  largely  used  fot 
this  purpose,  also  several  types  of  dry  cells. 


Fig.  6. 


ANNUNCIATOR  SYSTEM. 

The  wiring  diagram  for  a simple  annunciator  system  is 
shown  in  Fig.  1.  The  pushes  1,  2,  3,  etc.  are'  located  in  the 
various  rooms,  one  side  being  connected  to  the  battery  wire 
b,  and  the  other  to  the  leading 
wire  l in  communication  with  the 
annunciator  drop  corresponding 
to  that  room.  A battery  of  2 or  3 
Leclanche  cells  is  placed  at  B in 
any  convenient  location.  The  size 
of  wire  used  throughout  may  be 
No.  18  annunciator  wire. 

A return-call  system  is  illus- 
trated in  Fig.  2,  in  which  there  is 
one  battery  wire  6,  one  return  wire 
r,  and  one  leading  wire  l , I2,  etc. 
for  each  room.  The  upper  portion  of  the  annunciator  board 
Is  provided  with  the  usual  drops,  and  below  these  are  the 


244 


ELECTRICITY. 


return-call  pushes.  These  are  double-contact  buttons,  held 
normally  against  the  upper  contact  by  a spring.  When 
in  this  position,  the  closing  of  the  circuit  by  the  push  button 
in  any  room,  such  as  No.  4,  rings  the  office  bell  and  releases 
No.  4 drop,  the  path  of  the  current  in  this  case  being  from 
push  4 to  a-c-d-e-f-g-B-h-b 
back  to  the  push  button. 
On  the  return  signal  being 
made  by  pressing  the 
button  at  the  lower  part  of 
the  annunciator  board,  the 
office-bell  circuit  is  broken 
at  d,  and  a new  circuit 
formed  through  k as  fol- 
lows: From  the  battery  B 
t o g-m-r-n-o-a-c-k-p  t o 
battery,  the  room  bell 
being  in  this  circuit.  A 
general  fire-alarm  may  be 
added  to  this  system,  con- 
sisting of  an  automatic 
clockwork  apparatus  for 
closing  all  the  room-bell 
02,  circuits  at  once,  or  as  many 
at  a time  as  a battery  can 
ring.  When  this  system  is 
should  be  either  No.  14  or 


Fig.  2. 


installed,  the  battery  wire 
No.  16.  Four  or  five  Leclanche  cells  are  usually  required 
in  this  case. 

It  will  be  seen  that  the  connections  are  so  arranged  that 
the  room  bell  will  ring  when  the  push  in  that  room  is  pressed. 
If  this  be  not  desired,  a double-contact  push  may  be  substi- 
tuted, so  that  the  room-bell  circuit  is  broken  at  the  same 
time  that  the  circuit  is  made  through  the  annunciator.  This 
double  push  should  be  so  connected  that  the  circuit  is 
normally  complete  through  the  bell,  the  leading  wire  being 
connected  to  the  tongue  and  the  battery  wire  being  con- 
nected to  the  second  contact  point,  which  is  normally  out  of 
circuit. 


UNDERWRITERS’  RULES. 


245 


EXTRACT  FROM  THE  REGULATIONS  OF  THE  UNDER- 
WRITERS' ASSOCIATION. 

Incandescent  Wires.— Conducting  wires,  carried  over  or 
attached  to  buildings,  must  be  (a)  at  least  7 ft.  above  the 
highest  point  of  flat  roofs,  and  ( b ) 1 ft.  above  the  ridge  of 
pitch  roofs;  (c)  when  in  proximity  to  other  conductors  likely 
to  divert  any  portion  of  the  current,  they  must  be  protected 
by  guard  irons  or  wires,  or  a proper  additional  insulation,  as 
the  case  may  require. 

For  entering  buildings,  (a)  wires  writh  an  extra-heavy 
waterproof  insulation  must  be  used;  ( b ) they  must  be  pro- 
tected by  drip  loops;  (c)  also  protected  from  abrasion  by 
awning  frames;  ( d ) be  at-  least  6 in.  apart;  ( e ) the  holes 
through  which  they  pass  in  the  outer  w'alls  of  such  buildings 
must  be  bushed  with  a non-inflammable,  waterproof,  insu- 
lating tube,  and  (/)  should  slant  upward  toward  the  inside. 

(a)  Wires  must  never  be  left  exposed  to  mechanical 
injury,  or  to  disturbance  of  any  kind,  (b)  Wires  must  not 
be  fastened  by  metallic  staples,  (c)  When  wires  pass  through 
walls,  floors,  partitions,  timbers,  etc.,  glass  tubing,  or  so-called 
“ floor  insulators,”  or  other  moisture-proof,  non-inflammable 
insulating  tubing  must  be  used,  (d)  At  all  outlets  to  and 
from  cut-outs,  switches,  fixtures,  etc., wires  must  be  separated 
from  gas  pipes  or  parts  of  the  building  by  pofcelain,  glass,  or 
other  non-inflammable  insulating  tubing,  (e)  and  should  be 
left  in  such  a way  as  not  to  be  disturbed  by  the  plasterers. 
(/)  Wires  of  whatever  insulation  must  not  in  any  case  be 
taped,  or  otherwise  be  fastened,  to  gas  piping.  ( g ) If  no  gas 
pipes  are  installed  at  the  outlets,  an  approved  substantial 
support  must  be  provided  for  the  fixtures. 

In  crossing  any  metal  pipes,  or  any  other  conductor, 
(a)  wires  must  be  separated  from  the  same  by  an  air  space 
of  at  least  i in.,  where  possible,  and  (b)  so  arranged  that 
they  cannot  come  in  contact  with  each  other  by  accident, 
(c)  They  should  go  over  water  pipes,  wThere  possible,  so  that 
moisture  will  not  settle  on  the  wires. 

In  unfinished  lofts,  between  floors  and  ceilings,  in  par- 
titions, and  other  concealed  places,  wires  must  (a)  be  kept 
free  of  contact  with  the  building;  ( b ) be  supported  on  glass, 


246 


ELECTRICITY. 


porcelain,  or  other  non-combustible  insulators;  (c)  have  at 
least  1 in.  clear  air  space  surrounding  them;  ( d ) be  at  least 
10  in.  apart,  when  possible;  and  ( e ) should  be  run  singly  on 
separate  timbers  or  studding.  (/)  When  thus  run  in  per- 
fectly dry  places,  not  liable  to  be  exposed  to  moisture,  a wire 
having  simply  a non-combustible  insulation  may  be  used. 

Soft  rubber  tubing  is  not  desirable  as  an  insulator. 

Care  must  be  taken  that  the  wires  are  not  placed  above 
each  other  in  such  a manner  that  water  could  make  a cross- 
connection. 

On  all  loops  of  incandescent  circuits,  safety  catches  must 
be  used  on  both  sides  of  the  loop,  and  switches  on  such  loops 
should  be  double-poled. 

Wires  must  not  be  fished  (a)  for  any  great  distance,  and 
(6)  only  in  cases  where  the  inspector  can  satisfy  himself  that 
the  above  rules  have  been  complied  with,  (c)  Twin  wires 
must  never  be  employed  in  this  class  of  concealed  work. 

Dynamo  Machines.— Dynamo  machines  must  be  located  in 
dry  places,  not  exposed  to  flyings  or  easily  combustible 
material,  and  insulated  upon  wooden  foundations.  The 
machines  must  be  provided  with  devices  that  shall  be 
capable  of  controlling  any  changes  in  the  quantity  of  the 
current;  and  if  the  governors  are  not  automatic,  a competent 
person  must  be  in  attendance  near  the  machine  whenever  it 
is  in  operation. 

Each  machine  must  be  used  with  complete  wire  circuits; 
and  connections  of  wires  with  pipes,  or  the  use  of  circuits  in 
any  other  method,  are  absolutely  prohibited. 

The  whole  system  must  be  kept  insulated,  and  tested  every 
day  with  a magneto  for  ground  connections  in  ample  time 
before  lighting,  to  remedy  faults  of  insulation,  if  they  are 
discovered;  and  proper  testing  apparatus  must  in  each  case 
be  provided.  This  applies  to  both  central  station  and  isolated 
plants. 

Testing  circuits  for  grounds  with  a battery  and  bell  is  not 
considered  a reliable  test. 

Preference  is  given  to  switches  constructed  with  a lapping 
connection,  so  that  no  electric  arc  can  be  formed  at  the  switch 
when  it  is  changed;  otherwise  the  stands  of  switches,  where 


UNDERWRITERS’  RULES. 


247 


powerful  currents  are  used,  must  be  made  of  some  incom- 
bustible substances  that  will  withstand  the  heat  of  the  arc 
when  the  switch  is  changed. 

Motors.— Wires  for  motors  should  be  run  exactly  as  for 
lamps  on  similar  circuits. 

On  low-tension  circuits,  where  motors  are  run  in  multiple, 
safety  catches  must  be  used  on  each  side  of  the  circuit. 

On  high-tension  circuits  the  same  restrictions  apply  as  for 
arc  lamps,  and  suitable  cut-outs  must  be  provided. 

Motors  must  be  treated  as  dynamos  as  regards  insulation, 
flyings,  dampness,  etc. 

Note.— If  the  regulations  of  the  Underwriters’  Association 
are  not  followed  in  wiring  buildings,  the  wiring  is  liable  to 
be  condemned  by  the  Insurance  Inspectors  and  the  policy 
canceled. 

WIRE  TABLES. 


Weight  of  Underwriters’  Line  Wire,  Insulated. 


No.  B.  & S. 

Pounds  per  1,000 
Feet. 

Feet  per  Pound. 

0000 

800 

1.25 

000 

666 

. 1.50 

00 

500 

2.00 

0 

363 

2.75 

1 

313 

3.20 

2 

250 

4.00 

3 

200 

5.00 

4 

144 

6.9 

5 

125 

8.0 

6 

105 

9.5 

7 

87 

11.5 

8 

69 

14.5 

10 

50 

20.0 

12 

31 

32.0 

14 

22 

45.0 

16 

14 

70.0 

18 

11 

90.0 

248 


ELECTRICITY. 


Equivalent  Sectional  Area  of  Wires,  B.  & S.  Gauge. 


Gauge  No. 

No.  of  Wires. 
Gauge  No. 

No.  of  Wires. 
Gauge  No. 

No.  of  Wires. 
Gauge  No. 

No.  of  Wires. 
Gauge  No. 

No.  of  Wires. 
Gauge  No. 

No.  of  Wires. 
Gauge  No. 

Gauge  No. 
Gauge  No. 

0000 

2-  0 

CO 

1 

8-  6 

16-  9 

32-12 

64-15 

000 

2-  1 

4-  4 

8-  7 

16-10 

32-13 

64-16 

00 

2-  2 

4-  5 

8-  8 

16-11 

32-14 

64-17 

1 and  3 

0 

2-  3 

4-  6 

8-  9 

16-12 

32-15 

64-18 

2 and  3 

1 

2-  4 

4-  7 

8-10 

16-13 

32-16 

3 and  5 

2 

2-  5 

4-  8 

8-11 

16-14' 

32-17 

4 and  6 

3 

2-  6 

4-  9 

8-12 

16-15 

32-18 

5 and  7 

4 

2-  7 

4-10 

8-13 

16-16 

6 and  8 

5 

QO 

4-11 

8-14 

16-17 

7 and  9 

6 

2-  9 

4-12 

8-15 

16-18 

8 and  10 

7 

2-10 

4-13 

8-16 

9 and  11 

8 

2-11 

4-14 

8-17 

10  and  12 

9 

2-12 

4-15 

8-18 

11  and  13 

10 

2-13 

4-16 

12  and  14 

11 

2-14 

4-17 

13  and  15 

12 

2-15 

4-18 

14  and  16 

13 

2-16 

15  and  17 

14 

2-17  1 

16  and  18 

15 

2-18  1 

The  above  table  indicates  the  number  of  smaller  wires 
required  to  give  a sectional  area  equal  to  one  larger  size  wire, 
the  figures  between  the  horizontal  lines  corresponding  to  each 
other.  For  example:  It  requires  two  wires,  No.  0,  or  4 wires, 
No.  3,  etc.,  to  give  a sectional  area  equal  to  1 wire,  No.  0000. 
Again:  it  requires  two  wires,  No.  13,  or  4 wires,  No.  16;  or  2 
wires,  1 No.  12  plus  1 No.  14,  to  give  a sectional  area  equal 
to  1 No.  10. 


WIRE  TABLES. 


249 


Comparative  Sizes  of  Wires,  B.  & S.  and  Birmingham 

Gauges. 


Diameter.  Inches. 

B.  &S. 

Birmingham. 

.460 

0000 

.454 

0000 

.425 

000 

.4096 

000 

.380 

00 

.3648 

00 

.340 

0 

.3249 

0 

.3000 

1 

.2893 

1 

.284 

2 

.259 

3 

.2576 

2 

.238 

4 

.2294 

3 

.22 

5 

.2043 

4 

.203 

6 

.1819 

5 

.18 

7 

.165 

8 

.162 

6 

.148 

9 

.1443 

7 

.134 

10 

.1285 

8 

.12 

11 

.1144 

9 

.109 

12 

.1019 

10 

.095 

13 

.0907 

11 

.083 

14 

250 


ELECTRICITY. 


Comparative  Sizes  of  Wires,  B.  & S.  and  Birmingham 
Gauges— ( Continued) . 


Diameter,  Inches. 

B.  & S. 

Birmingham. 

.0808 

12 

.0720 

13 

15 

.0650 

16 

.0641 

14 

.0580 

17 

.0571 

15 

.0508 

16 

.0490 

18 

.0453 

17 

.0420 

19 

.0403 

18 

.0359 

19 

Note.— B.  & S.  gauge  is  generally  used  in  America. 


Comparison  of  Properties  of  Aluminum  and  Copper. 


Aluminum. 

Copper. 

Conductivity  (for  equal  sizes) 

.54  to  .63 

1. 

Weight  (for  equal  sizes) 

Weight  (for  equal  length  and  re- 

.33 

1. 

sistance)  

Price  (per  pound)  Aluminum,  29c.; 

.48 

1. 

Copper,  16c.  (bare  wire)  

Price  (equal  length  and  resistance, 

1.81 

1. 

bare  line  wire) 

Temperature  coefficient  per  de- 

'.868 

1. 

gree  F 

.002138 

.002155 

Resistance  of  mil-foot  (20°  C.) 

18.73 

10.5 

Specific  gravity  

2.5  to  2.68 

8.89  to  8.93 

Breaking  strength  (equal  sizes)  

1. 

1. 

WIRE  TABLES. 


251 


Resistance  of  Pure  Copper  Wire. 


OQ 

M 

Resistance  at  75°  F. 

R.  Ohms  per 

Ohms  per 

Feet  per 

Ohms  per 

1,000  Feet. 

Mile. 

Ohm. 

Pound. 

4-0 

.04904 

.25891 

20,392.90 

.00007653 

3-0 

.06184 

.32619 

16,172.10 

o00012169 

00 

.07797 

.41168 

12,825.40 

.00019438 

0 

.09827 

.51885 

10,176.40 

.00030734 

1 

.12398 

.65460 

8,066.00 

.00048920 

2 

.15633 

.82513 

6,396.70 

.00077784 

3 

.19714 

1.04090 

5,072.50 

.00123700 

4 

.24858 

1.31248 

4,022.90 

.00196660 

5 

.31346 

1.65507 

3,190.20 

.00312730 

6 

.39528 

2.08706 

2,529.90 

.00497280 

7 

.49845 

2.63184 

2,006.20 

.00790780 

8 

.62849 

3.31843 

1,591.10 

.01257190 

9 

.79242 

4.18400 

1,262.00 

.01998530 

10 

.99948 

5.27726 

1,000.50 

.03170460 

11 

1.26020 

6.65357 

793.56 

.05054130 

12 

1.58900 

8.39001 

629.32 

.08036410 

13 

2.00370 

10.57980 

499.06 

.12778800 

14 

2.52660 

13.34050 

395.79 

.20318000 

15 

3.18600 

16.82230 

313.87 

.32307900 

16 

4.01760 

21.21300 

248.90 

.51373700 

17 

5.06600 

26.74850 

197.39 

.81683900 

18 

6.38800 

33.72850 

156.54 

1.29876400 

19 

8.05550 

42.53290 

124.14 

2.06531200 

20 

10.15840 

53.63620 

98.44 

3.28437400 

252 


ELECTRICITY. 


DYNAMOS  AND  MOTORS. 


253 


In  the  diagrams  showing  the  connections  of  dynamo- 
electric  machines,  the  heavy  coils  represent  the  series  wind- 
ing on  the  field  magnets  through  which  the  entire  current  of 
the  machine  passes;  the  lighter  coils  represent  the  shunt 
winding  on  the  field  magnets  through  which  only  part  of 
the  main  current  passes. 

Lamps  connected  in  series. 


Lamps  connected  in  multiple- 
arc  or  parallel. 


Edison  three-wire  system. 


DYNAMOS  AND  MOTORS. 


MOTOR  CIRCUITS. 

To  find  the  size  of  wire  on  stationary  motor  circuits: 

Let  c.  m.  = circular  mils; 

e = E.  M.  F.  of  motor  in  volts; 
v = loss  of  volts  in  line; 
d = distance  from  generator  to  motor  in  feet; 
k = efficiency  of  motor; 

10.8  ohms  is  the  resistance  of  1 ft.  of  commercial  copper 
wire  1 mil  in  diameter. 

_ H.  P.  of  motor  X 746  X 2 d X 10-8 
evk 

Approximate  Motor  Efficiency. 

£ to  1£  H.  P.  inclusive  = 75 $ efficiency. 

3 to  5 H.  P.  inclusive  = 80$  efficiency. 

7£  to  10  H.  P.  inclusive  = 85$  efficiency. 

15  H.  P.  and  upwards  = 90$  efficiency. 


DSC  620IM8M 

620IH8M  INTERNftT  I OHfiL  CORRESPONDED 

MECHANICS’  POCKET  MEH0RANBR$7TH  EB#& 
69879?  1904  ? ADDED: 

01  001  314  Em 

0?  00?  3W  Em 

PAGE  1 END 


: SCHOOLS.  SCRhHTOM 
i’HHTOH  4-33907 
'780603 


254 


ELECTRICITY. 


Under  ordinary  circumstances,  10$  loss  from  generator  to 
motor  is  a maximum  on  stationary  motor  circuits. 

Example. — What  is  the  size  of  wire  necessary  for  a circuit 
on  which  a 10  H.  P.  500-volt  motor  is  running,  when  the  dis- 
tance between  the  motor  and  generator  is  2,000  ft.  and  the 
loss  is  5$  ? 

Solution.— Volts  at  generator,  ^ = 526,  nearly. 


Volts  lost  in  line,  526  — 500  = 26. 

In  the  table  on  page  253,  the  approximate  efficiency  of  a 
10  H.  P.  motor  is  given  as  85$. 

10  X 746  X 4,000  X 10.8 
C'  m‘  - 600X26X^86  29’165' 

In  the  table  on  page  238,  the  nearest  size  of  wire  corre- 
sponding to  this  area  is  No.  6 B.  & S.  gauge. 

The  approximate  weight  and  resistance  per  mile  of  round 
bare  wire  when  d is  the  diameter  in  mils,  are,  for  copper  wire, 

d 2 „ , 56,970  . . . d2  380,060 

lb.  and  -^2—  ohms;  for  iron  wire,  lb.  and  — 

ohms. 

Copper  wire  is  approximately  If  times  the  weight  of  an 
iron  wire  of  the  same  diameter. 

In  determining  the  size  of  wire  to  be  used  for  inside  work, 
after  finding  the  c.  m.,  always  refer  to  the  table  on  page  238, 
and  see  that  the  wire  obtained  by  the  formula  is  sufficiently 
large  to  carry  the  current;  if  not,  use  larger  wire,  regardless 
of  per-cent.  loss.  For  pole-line  construction , never  use  wire 
smaller  than  No.  V ;-*>  & S.  gauge . 


DYNAMO  DESIGN. 

The  fundamental  principle  of  dynamo  design  is  expressed 


by  the  formula 


_ NCn 
* 108  X 60’ 


in  which 

E = electromotive  force  in  volts  given  by  the  dynamo; 

N = number  of  lines  of  force  used  to  magnetize  the  armature; 
C = number  of  conductors  in  a bipolar  machine,  measured 
all  round  the  outside  of  the  armature  (whether  in  one 


DYNAMO  DESIGN. 


255 


or  more  layers),  or  in  a multipolar  machine,  as  measured 
from  a point  opposite  one  north  pole  to  a corresponding 
point  opposite  the  next  succeeding  north  pole; 
n = number  of  revolutions  per  minute  of  the  armature. 

For  example,  a 2-pole  dynamo  has  2,000,000  lines  of  force 
passing  from  the  north  pole  through  the  armature  to  the 
south  pole;  there  are  200  conductors  on  the  surface  of  the 
armature,  and  the  speed  is  1,500  rev.  per  min.  The  electro- 
motive force  generated  will  then  be 


E = 


2,000,000  X 200  X 1,500 
100,000,000  X 60 


= 100  volts. 


If  a 4-pole  dynamo  were  used,  having  a 4-circuit  armature 
and  4 sets  of  brushes,  with  1,000,000  lines  of  force  passing 
through  any  one  pole  piece,  then  the  total  number  would  be 
2,000,000,  because  the  same  lines  of  force  pass  into  a south 
pole  that  emerge  from  a north  pole.  With  the  same  arma- 
ture as  above,  the  number  of  conductors  to  be  counted  is 
only  100,  as  taken  from  one  north  pole  to  the  next,  and  the 
electromotive  force  is 


E = 


2,000,000  X 100  X 1,500 
100,000,000  X 60 


= 50  volts. 


For  determining  the  number  of  lines  of  force  required 
in  a specific  case,  the  above  formula  may  be  reversed,  and  we 
have  AT  EX  108 X 60 

These  lines  of  force  have  a cir- 
cuit to  traverse  composed  of  three 
different  paths.  One  of  these  is 
through  the  field  magnet  and  yoke 
M,  Fig.  1;  next,  through  a double 
air  gap  (?;  and,  lastly,  through  the 
armature  core  A.  A given  density 
of  lines  of  force  may  not  be  ex- 
ceeded, this  limit  being  for  ordi- 
nary cast  iron  about  50,000  lines 
per  square  inch;  for  wrought-iron 
forgings  or  cast  steel,  about  90,000; 
and  for  soft  sheet  iron,  110,000. 

The  ratio  of  magnetization  to  magnetizing  force  is  called 


Ones  of  Force  per  Square  lnok» 


256 


ELECTRICITY. 


Fig.  2. 


DYNAMO  DESIGN. 


257 


the  'permeability.  The  permeability  of  air  is  very  low,  the 
intensity  of  magnetization  being  a direct  measure  of  the 
magnetizing  force  required;  therefore,  the  air  gap  is  usually 
made  short. 

In  order  to  drive  the  lines  of  force  through  the  magnetic 
circuit,  magnetizing  coils  are  wound  on  the  cores  at  M,  M.  A 
certain  number  of  ampere-turns  will  be  required,  depending 
on  the  density  of  the  lines  of  force  and  the  permeability  of  the 
different  portions  of  the  circuit.  The  number  of  turns  may  be 
found  by  taking  a convenient  current  value,  and  dividing  the 
ampere-turns  by  this.  Reference  to  a wire  table  will  then 
determine  whether  the  resistance  of  the  wi*e  will  be  such  that 
the  terminal  E.  M.  F.  of  the  machine  will  supply  the  proper 
current.  A margin  should  be  allowed  for  regulating,  and  for 
the  increase  in  resistance  due  to  rise  in  temperature,  which  is 
about  A<fo  for  every  degree  centigrade,  or  .222 <fo  for  every  degree 
Fahrenheit  above  75°  F. 

In  the  saturation  curves  of  Fig.  2 are  represented  graphic- 
ally the  different  values  of  the  induction  (B)  in  lines  of  force 
per  square  inch,  corresponding  to  the  magnetizing  force 
expressed  in  ampere-turns  per  inch  of  length  of  circuit.  Thus, 
to  send  70,000  lines  of  force  through  a cast-steel  core  1 sq.  in.  in 
cross-sectional  area,  would  require  about  30  ampere-turns  for 
every  inch  in  length  of  core.  The  30  ampere-turns  might  be 
obtained  by  using  a coil  of  30  turns  carrying  1 ampere,  or  300 
turns  of  yq  ampere,  etc.  The  number  of  lines  of  force  N for 
any  particular  case  being  known,  and  also  the  allowable 
density  B,  which  will  vary  somewhat  with  different  samples 

jV 

of  iron,  the  cross-sectional  area  A = 

’ B 

The  ampere-turns  to  be  added  to  the  magnetizing  coils  to 
overcome  the  resistance  of  the  air  gap  is 


where  H = number  of  lines  of  force  per  square  inch; 
and  l = length  of  air  gap  (the  two  sides  added  together) 
in  inches,  usually  a fraction. 

It  is  necessary,  in  calculating  the  ampere-turns  for  the  field 
circuit,  to  allow  for  leakage  of  lines  of  force  through  the 


258 


ELECTRICITY. 


surrounding  air,  as  the  total  number  generated  does  n^t 
pass  through  the  armature  core.  This  leakage  may  amount 
to  30 $ or  40$  of  the  whole,  but  is  much  less  in  well-designed 
machines. 

For  example,  a bipolar  dynamo  has  magnet  cores  having  a 
mean  length,  with  pole  pieces,  of  10  in.  each;  the  yoke  of  the 
magnet  is  13  in.;  air  gap,  x%  in.  each  side;  armature  core, 
10  in.  The  magnetic  density  in  the  core  is  85,000;  air  gap, 
46,000;  yoke,  65,000;  armature  core,  90,000  lines  of  force  per 
square  inch.  If  the  fields  are  wrought-iron  forgings,  and  the 
armature  is  built  up  of  soft  sheet  iron,  then  the  ampere-turns 
necessary  will  be* 

A.-T.  Ampere- 

Length.  B per  In.  Turns. 


Magnet  cores 20  in.  85,000  44  880  * 

Yoke 13  in.  65,000  16  208 

Armature 10  in.  90,000  40  400 

Air  gap % in.  46,000  5,425 


Total  ampere-turns  6,913 

In  determining  the  size  of  wire  to  be  used  in  the  armature 
winding,  a certain  density  of  current  may  be  assumed  as  the 
limit.  This  is  usually  expressed  in  circular  mils  or  thou- 
sandths Ox  an  inch  per  ampere.  For  most  purposes  of  design, 
a density  of  600  circular  mils  per  ampere  may  be  allowed. 
In  estimating  the  current  passing  through  the  armature,  it 
must  be  remembered  that  the  current  of  the  outside  circuit 
divides  on  reaching  the  armature,  and  passes  through  it 
along  two  paths  in  parallel  with  each  other. 


FAULTS  OF  DYNAMOS. 

Reversal  of  Field.— Run  the  machine  up  to  speed,  and  hold 
a small  compass  near  each  pole  piece  in  succession.  Their 
polarity  should  alternate  all  the  way  round. 

Failure  to  Build  Up.— This  is  probably  due  to  reversal  of 
shunt  connections.  Rock  the  brushes  around  until  any  one 
set  occupies  a position  formerly  occupied  by  the  next  set.  If 
this  should  remedy  the  trouble,  and  such  position  is  incon- 
venient, move  them  back  and  reverse  connections  of  shunt 


DYNAMO  DESIGN. 


259 


windings.  If  the  failure  of  machine  is  due  to  want  of 
residual  magnetism,  send  a current  from  some  external 
source  through  the  fields.  If  it  is  due  to  a broken  circuit, 
each  coil  may  be  tested  separately  with  a battery  and 
galvanometer  or  low-reading  Weston  voltmeter.  Failure 
to  generate  may  be  due  to  the  brushes  being  out  of  the  neutral 
plane,  which  may  be  tested  by  moving  them  into  different 
positions. 

Heating.— This  maybe  caused  by  a short-circuited  armature 
coil.  Allow  the  machine  to  cool,  then  run  for  a few  minutes 
with  no  load,  and  stop.  The  defective  coil  will  be  found  to 
be  much  hotter  than  the  rest.  It  should  be  marked,  and  the 
armature  taken  out,  when  the  coil  may  be  rewound  or 
otherwise  repaired.  If  the  heating  is  even,  the  load  may  be 
excessive  and  should  be  reduced.  The  effect  may  be  due  to 
eddy  currents  in  the  armature  core,  but  this  is  a question  of 
design  in  the  first  instance. 

Sparking  at  Commutator.— If  this  be  due  to  overload,  the 
sparking  cannot  be  cured  except  by  reducing  the  load.  The 
trouble  may  be  due  to  improper  position  of  brushes.  Move 
the  rocker-arm  to  one  side  or  the  other  to  determine  this. 
If  copper  brushes  (tangential)  are  used,  they  may  be 
unevenly  spaced  round  the  commutator;  each  set  of  brushes 
should  have  the  same  relative  position  with  regard  to  the 
respective  pole  tips.  Sparking  may  be  caused  by  an  uneven 
commutator,  in  which  case  it  should  be  smoothed  with 
sandpaper  (never  emery)  or  turned  down  in  the  lathe. 
A broken  connection  at  the  commutator  leads  will  produce 
flashing  at  each  revolution,  and  one  of  the  bars  will  show  a 
burn  extending  nearly  across  it.  The  loose  wire  should  be 
secured,  or  if  broken,  the  commutator  bars  may  be  connected 
together  with  a piece  of  wire  or  a drop  of  solder  as  a 
temporary  repair.  As  soon  as  possible  a new  coil  should  be 
put  in.  Sparking  may  also  occur,  in  a multipolar  machine, 
from  the  wearing  away  of  the  bearings,  which  produces 
eccentricity  of  the  armature  with  respect  to  field,  and  conse- 
quent unequal  magnetic  induction  at  different  points. 
A slight  sparking  at  the  brushes  of  the  machine  is  not 
detrimental. 


260 


ELECTRICITY. 


OUTPUT  AND  EFFICIENCY  OF  MOTORS. 

A dynamo,  when  supplied  with  current  from  an  external 
source,  becomes  a motor,  turning  the  electrical  energy  into 
mechanical  energy.  The  ratio  between  these  two  quanti- 
ties, that  is,  between  the  input  and  output,  determines  the 
efficiency  of  the  motor.  The  input  may  be  found  by  measur- 
ing the  current  C with  an  ammeter,  and  the  voltage  E with 
a voltmeter,  their  product  giving  the  power  supplied  in  watts, 
W = CE.  This  quantity,  divided  by  746,  gives  the  electrical 
W 

horsepower,  or  E.  H.  P.  = 


The  output  is  measured  by  means  of  a Prony  brake  (see 
figure).  The  motor  pulley  P is  clamped  between  two  blocks 


of  wood  B,  B , their  pressure  being  regulated  by  the  thumb- 
screws N,  JV,  on  the  long  bolts  which  hold  them  together. 
The  lower  block  is  extended  to  form  an  arm  A of  convenient 
length,  and  furnished  with  a sharp  lagscrew  C at  the  end,. 
The  lagscrew  presses  on  the  platform  of  a set  of  scales  S, 
whereby  its  pressure  may  be  determined.  A counterbalance 
at  W neutralizes  the  weight  of  the  arm.  When  the  pulley  is 
revolved  in  the  direction  shown,  the  pressure  on  the  scale 
will  indicate  the  torque,  or  twisting  power,  developed,  which 


ELECTRIC  MOTORS. 


261 


is  expressed  as  the  product  of  the  pressure  on  the  scale  into 
the  distance  between  the  center  of  pulley  and  the  point  of 
the  screw.  If  the  length  of  arm  E = 2 ft.,  and  the  pressure 
is  50  lb.,  the  torque  T = 100  ft. -lb.  The  horsepower  may  be 
determined  by  the  following  formula: 

2 X 3.1416  TS 
‘ * 33,000  ’ 

in  which  5 is  the  speed  of  motor  in  revolutions  per  minute. 


APPLICATIONS  OF  ELECTRIC  MOTORS. 

The  same  varieties  of  field  and  armature  connections  are 
used  for  motors  as  for  dynamos,  namely,  series,  shunt,  and  com- 
pound, and  each  type  has  distinguishing  characteristics.  The 
series  motor  is  especially  suitable  for  use  in  cases  wrhere  a very 
high  starting  torque  is  required  in  order  to  obtain  rapid 
acceleration  under  load,  as,  for  instance,  in  street-railway 
work.  Torque  may  be  defined  as  the  reaction  of  the  current 
in  the  armature  or  moving  part  against  the  magnetic  lines  of 
force  in  the  field  magnets  or  stationary  part.  Strength  of 
field  is  obtained  by  the  current  circulating  through  the  mag- 
net coils;  consequently,  the  torque  in  a series  motor  will  be  a 
maximum  when  the  current  passing  through  is  a maximum, 
as  the  same  amount  flows  through  armature  and  field.  The 
opposition  to  the  flow  of  current  is  the  resistance  of  the  cir- 
cuit and  the  counter  E.  M.  F.  of  the  armature.  When  the 
current  is  applied,  its  value  is  determinable  by  Ohm’s  law 
for  the  first  moment,  supposing  self-induction  to  be  eliminated. 
The  resistance  of  a series  motor  is  usually  so  low  that  an 
additional  resistance  must  be  used  at  starting  in  order  to 
prevent  an  excessive  flow  of  current;  but,  as  soon  as  the 
armature  begins  to  revolve,  the  counter  E.  M.  F.  opposes  and 
cuts  down  the  current,  and,  consequently,  the  torque.  The 
speed  will  continue  to  increase  and  the  torque  to  decrease 
until  the  mechanical  resistance  to  rotation  balances  the 
torque.  If  the  motor  is  running  light,  the  speed  will  rise 
continually,  the  counter  E.  M.  F.  will  also  increase  and  cut 
down  the  current,  and  the  consequent  reduction  of  field 
strength  will  require  a still  higher  speed  in  order  to  develop 


262 


ELECTRICITY 


the  necessary  counter  E.  M.  F.,  the  final  result  being, 
probably,  the  bursting  of  the  armature.  The  speed  of  a 
series  motor  under  a constant  load  may  be  regulated  by  the 
somewhat  wasteful  method  of  introducing  a resistance  in 
series  to  reduce  the  speed,  and  by  cutting  out  or  shunting 
part  of  the  field  coils,  to  increase  it.  When  two  motors  are 
used,  they  may  be  put  in  series  at  starting  and  connected  in 
parallel  for  higher  speeds.  The  series  motor  is  well  adapted 
for  electric  cranes,  because  it  will  automatically  regulate  its 
speed  to  the  weight  to  be  raised,  exerting  a very  powerful 
torque  at  low  speed  for  a heavy  load. 

The  shunt  motor  will  give  a nearly  constant  speed  for  any 
variation  in  load,  as  long  as  the  potential  of  current  supply 
(the  applied  E.  M.  F.)  is  constant.  This  condition  produces 
a constant  field,  as  the  shunt  winding  is  directly  across  the 
main  leads,  and  the  speed  of  the  motor  will  then  be  such 
that  the  difference  between  the  E.  M.  F.  of  supply  and  the 
counter  E.  M.  F.,  divided  by  the  resistance  of  the  armature, 
will  be  equal  to  the  current  passing  through  the  armature. 
A change  in  the  current  will  then  produce  but  a relatively 
small  change  in  the  required  counter  E.  M.  F.  of  the  motor, 
and  the  speed  will  only  vary  to  that  extent.  As  the  load  is 
put  on,  the  motor  tends  to  slow  down;  but  this,  by  decreasing 
the  counter  E.  M.  F.,  allows  more  current  to  flow,  thereby 
producing  more  torque  to  overcome  the  added  mechanical 
resistance.  Change  of  speed  may  be  produced  by  varying 
the  strength  of  the  magnetic  field,  the  weaker  the  field  the 
higher  the  speed.  If  the  load  is  constant,  the  torque  will  be 
decreased,  but,  if  the  load  be  correspondingly  increased,  the 
torque  will  remain  nearly  constant.  Considerable  weakening 
of  the  field  is  inadvisable,  as  it  will  cause  destructive  spark- 
ing at  the  commutator.  The  theoretically  perfect  method  of 
speed  regulation  for  a shunt  motor  is  to  provide  a constant 
and  independent  field,  and  effect  change  of  speed  by  varying 
the  applied  E.  M.  F.  at  the  armature  terminals  without 
insertion  of  extra  resistance.  In  this  case  the  torque  will 
always  be  proportional  to  the  load,  and  the  efficiency  will  be 
constant  and  independent  of  speed  and  torque.  In  the 
operation  of  such  a system,  certain  complications  are  intro- 


BATTERIES. 


263 


duced,  inasmuch  as  it  is  necessary  to  install  in  connection 
with  each  motor  a special  dynamo  with  variable  field,  and 
this  condition  may  therefore  constitute  a serious  objection 
when  the  first  cost  of  the  plant  is  required  to  be  low. 

A differential  compound  winding  may  be  used  when  a 
more  nearly  constant  speed  is  wanted.  The  series  turns  on 
the  field  magnets  are  so  connected  as  to  oppose  the  shunt 
turns,  and  when  an  increase  of  load  tends  to  cut  down 
the  speed,  the  additional  current  through  the  series  turns 
weakens  the  field  slightly,  so  that  the  same  speed  as  before 
is  required  to  generate  the  lower  counter  E.  M.  F. 

Shunt  motors  are  especially  useful  for  machine  tools, 
which  require  a constant  speed  irrespective  of  load,  and 
may  also  be  used  on  printing  presses  and  similar  machines 
where  the  load  is  more  nearly  uniform.  When  a variation 
in  speed  with  load  is  immaterial,  a cumulative  compound 
winding  may  be  employed,  in  which  the  series  turns  act 
with  the  shunt,  thereby  increasing  the  torque  at  starting, 
and  affording  some  of  the  characteristics  of  both  the  shunt 
and  series  windings. 


BATTERIES. 

The  simple  primary  battery  consists  of  two  elements,  the 
anode , which  is  usually  zinc,  and  the  cathode , which  may  be 
carbon,  both  immersed  in  an  exciting  liquid  called  the 
electrolyte.  The  chemical  action  incident  to  the  generation 
of  current  dissolves  the  zinc  and  liberates  free  hydrogen  at 
the  cathode,  which  adheres  to  the  surface  and  reduces  the 
E.  M.  F.  of  the  battery.  To  overcome  this  effect,  called 
polarization , a depolarizer  is  used  which  will  take  up  the 
hydrogen  as  it  is  formed. 

Depolarizers  may  be  solid  or  liquid.  When  solid,  the 
material  is  usually  packed  round  the  cathode,  as  in  the 
case  of  the  Leclanch6  cell;  when  the  depolarizer  is  liquid,  it 
may  be  prevented  from  mixing  with  the  electrolyte  by  a 
porous  partition,  or,  if  their  specific  gravities  differ  consider- 
ably, they  will  remain  separated  one  over  the  other  in  the 
jar.  The  following  table  gives  the  elements  and  depolarizers 
for  different  cells,  with  the  E.  M.  F.  in  volts: 


264 


ELECTRICITY. 


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BATTERIES. 


265 


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266  ELECTRICITY. 


STORAGE  BATTERIES. 


267 


STORAGE  BATTERIES. 

Storage  batteries  or  accumulators  are  composed  of  plates 
of  prepared  lead,  placed  side  by  side  in  glass  cells  or  wooden 
boxes  lined  with  rubber  or  lead,  alternate  plates  being  con- 
nected together,  thus  forming  two  sets,  which  constitute  the 
positive  and  negative  elements.  The  plates  are  entirely  sub- 
merged in  dilute  sulphuric  acid,  specific  gravity  1.17.  The 
charging  E.  M.  F.  is  about  2.5  volts  per  cell,  so  that,  if  10  cells 
are  connected  in  series,  the  required  E.  M.  F.  will  be  25  volts. 
The  discharging  E.  M.  F.  is  usually  taken  as  1.9  volts,  so  that 
an  installation  to  supply  current  at  115  volts  should  consist  of 
115 

— = 61  cells,  with  a few  added  to  replace  any  that  are  out 

of  order  or  to  serve  as  regulators  to  vary  the  E.  M.  F.  As  soon 
as  the  battery  is  set  up  and  the  electrolyte  added,  the  charging 
should  commence,  the  first  charge  being  continued  a long 
while  at  a comparatively  slow  rate.  Observe  that  the  direc- 
tion of  current  through  the  cell  in  charging  is  from  the  positive 
or  brown  plate  to  the  negative  or  gray  one.  Discharging 
should  be  at  a low  rate,  as  rapid  discharge  leads  to  deteriora- 
tion of  the  positive  plates. 

The  rating  of  the  capacity  of  accumulators  is  usually  made 
on  the  basis  of  a discharge  current  that  will  cause  the  E.  M.  F. 
to  fall  to  1.8  volts  in  10  hours,  but  it  is  well  to  stop  discharging 
when  the  E.  M.  F.  falls  to  1.9  volts. 

Storage-Battery  Regulation.— -In  electric-lighting  plants,  an 
equalization  of  load  on  the  dynamos  is  sometimes  obtained 
by  installing  accumulators  or  storage  batteries.  Automatic 
or  hand  regulation  may  be  employed,  the  usual  method  being 
to  cut  out  one  or  more  cells  when  the  load  is  light  and  change 
the  remainder,  these  cells  being  connected  in  again  when 
the  load  rises.  The  following  method  obviates  the  many 
disadvantages  of  this  system. 

A shunt  dynamo  d,  Fig.  1,  supplies  current  to  the  lighting 
mains  m,  n,  this  current  passing  through  the  fields  c of  a low- 
voltage  dynamo  or  booster  6,  driven  by  a shunt  motor  and  con- 
nected across  the  mains  in  series  with  the  battery  B.  The 
E.  M.  F.  of  the  dynamo  d is  a little  greater  than  that  of  the 
battery,  so  that  it  will  charge  the  battery  when  there  is  no 


268 


ELECTRICITY. 


Sa. 


external  load.  When  all  the  lights  are  turned  on,  the  booster 
field  will  be  fully  energized,  and  the  E.  M.  F.  of  the  booster 
will  be  added  to  that  of  the  bat- 
tery, thereby  causing  the  battery 
to  discharge  and  assist  the  dy- 
namo. At  a medium  load,  the 
battery  will  be  neutral,  neither 
taking  current  nor  discharging, 
while  the  dynamo  is  running 
at  full  load.  Any  increase  that 
may  be  made  in  the  load  will  then  be  taken  up  by  the 
battery. 

In  electric-railway  plants  the  dynamos  are  usually  over- 
compounded, thus  giving  a higher  E.  M.  F.  at  the  brushes 
at  full  load  than  at  light  load.  In  a case  of  this  kind,  a 
differential  winding  is  employed,  as  shown  in  Fig.  2,  which 


Fig.  1. 


causes  the  booster  to  work  both  ways.  On  light  loads 
a differential  winding  will  assist  the  dynamos  d'  and  d"  to 
charge  the  battery,  raising  the  E.  M.  F.  to  the  required 
value;  but  on  heavy  loads  the  series  winding  c will  over- 
power the  shunt  s,  and  the  battery  will  discharge  into  the 
outer  circuit.  The  shunt  field  must  be  regulated  so  that  the 
total  charging  and  discharging  that  is  done  within  a given 
time  will  balance  each  other,  as  the  battery  will  otherwise 
tend  either  to  overcharge  or  to  undercharge.  If  the  shunt 
field  is  strengthened,  it  will  cause  the  batteries  to  charge, 
while  if  the  field  is  weakened,  it  will  cause  the  batteries  to 
discharge  at  a lower  value  of  the  external  load  than 
before. 


ELECTRIC  GAS  LIGHTING 


269 


ELECTRODEPOSITION. 

For  electrodeposition  of  metals,  low-resistance  primary  bat- 
teries giving  from  2 to  10  volts  may  be  used  when  the  work 
is  on  a small  scale.  For  larger  work,  accumulators  may  be 
employed,  or  the  current  may  be  taken  directly  from  a low- 
voltage  dynamo.  The  electroplating  bath  consists  of  a solu- 
tion that  has  little  or  no  chemical  action  on  the  objects  to  be 
plated,  and  that  are  suspended  in  it  and  electrically  connected 
to  the  negative  pole  of  the  battery.  The  anode  is  a plate  of 
the  metal  that  it  is  desired  to  transfer;  it  is  also  submerged  in 
the  solution  and  connected  through  a resistance,  if  necessary, 
to  the  positive  pole  of  the  battery.  For  deposition  of  copper, 
the  bath  is  made  by  taking  4 parts  saturated  solution  of  sul- 
phate of  copper  mixed  with  1 part  of  water  containing  one- 
tenth  its  volume  of  sulphuric  acid.  The  current  used  must 
not  exceed  18  amperes  per  square  foot  of  surface  of  cathode. 
For  nickel,  use  the  double  sulphate  of  nickel  and  ammonia, 
specific  gravity  1.03;  the  current  density  must  be  low,  and 
the  solution  should  be  neutral  or  slightly  alkaline,  as  an  acid 
bath  will  cause  the  nickel  to  peel  off.  For  silver,  the  bath  is 
a solution  of  cyanide  of  silver  dissolved  in  cyanide  of  potas- 
sium. For  gold,  use  cyanide  of  gold  dissolved  in  cyanide  of 
potassium.  This  solution  is  kept  at  150°  F.  while  in  use. 


ELECTRIC  GAS  LIGHTING. 

The  arrangement  of  the  apparatus  required  for  electric 
gas  lighting  is  shown  in  the  figure,  A battery  of  about  6 


Leclanch6  cells  c,  c,  etc.,  joined  up  in  series,  is  connected  to 
one  terminal  of  a spark  coil  k,  the  other  terminal  of  which  is 
soldered  to  a gas  pipe  p.  The  wire  from  the  free  end  of  the 


270 


ELECTRICITY. 


battery  is  carried  up  through  the  house,  and  branches  are 
run  to  the  burners  as  at  5,  wherever  needed.  The  insulation 
of  this  wire  must  be  very  thorough,  special  precautions  being 
taken  when  it  is  carried  through  or  along  the  fixtures.  The 
burners  are  provided  with  a chain  a attached  to  a movable 
contact  spring,  which  is  drawn  past  the  burner,  producing  a 
spark  of  sufficient  intensity  to  ignite  the  gas  if  it  is  previously 
turned  on. 

In  multiple  gas  lighting,  a fine  wire  is  run  from  one  burner 
to  another  of  a group,  as  on  a chandelier,  leaving  a small  air 
gap  at  each  one,  and  a current  of  very  high  tension  is  used, 
generated  by  a small  frictional  machine,  causing  a spark  at 
each  burner.  The  last  contact  in  a series  of  burners  is  con- 
nected to  the  gas  pipe. 


THE  WHEATSTONE  BRIDGE. 

A diagrammatic  sketch  of  the  Wheatstone  bridge  is  shown 
in  Fig.  1.  This  instrument  is  widely  used  for  the  determina- 
tion of  unknown  resistances,  and  consists  of  such  an  arrange- 
ment of  three  circuits,  M,  N,  P,  of  variable  resistance,  that 

the  value  of  a fourth 
may  be  found  from 
their  relation.  This 
unknown  resistance 
is  connected  between 
the  points  b and  c, 
and  the  battery  B be- 
tween a and  b.  The 
variable  resistances 
are  then  so  adjusted 
that  there  shall  be  no 
difference  of  potential  between  c and  d , which  form  the  termi- 
nals of  the  galvanometer  G.  The  drop  in  potential  from  a to 
c will  then  be  the  same  as  from  a to  d,  and  a c bears  the  same 
proportion  to  a c b as  a d bears  to  adb.  From  this  it  follows 

MP 


Fig.  1. 


that  ac:  ad  = cb  : db,  or  the  unknown  resistance  X — 

M 10 

For  a certain  test,  the  ratio  of  the  arms,  — = — — . 

N lOO 


N ' 


On 


CABLE  TESTING. 


271 


adjusting  the  resistance  P,  a balance  is  obtained  when  it  is 
equal  to  7,800  ohms.  Then, 

„ 10  X 7,800  _OA  , 

X = — • - = 780  ohms. 

100 

A commercial  form  of  bridge  is  shown  in  Fig.  2.  The  same 
letters  of  reference  are  used  as  in  the  preceding  diagram. 
Two  keys,  K and  K',  are  added,  to  be  used  in  closing  the 


r 


u 

K’  \ 


jOC 


K 


>_j  A)  I 

“ • <1 


10  too  1000 


]CDCjt 


lOOOlOO  10 


)OGDC 


1 2 2 J 5 10  20  20 

> 

-OOGDaODCX 


3000  2000  1000  5 OO  200  200  LOO  50 


Fig.  2. 


<1 

.P  g 


si 


circuits.  Resistances  are  put  in  by  withdrawing  the  plugs. 
In  the  arm  JV  there  is  a resistance  of  10  ohms;  in  M,  1,000 
ohms;  in  P,  5,838  ohms.  If  the  galvanometer  G indicates  a 
balance,  the  value  of  the  unknown  resistance 


X = 


1,000  x 5,838  coo  OAA  , 

— Jq-- — = 583,800  ohms. 


CABLE  TESTING. 

Test  for  Capacity.— A condenser  of  known  capacity  k is 
charged  by  a battery  and  discharged  through  a galvanometer, 
producing  a deflection  c^.  The  cable,  having  an  unknown 
capacity  fc2,  is  charged  and  discharged  in  similar  manner, 

giving  a deflection  d2.  Then  k2  = ki  The  connections 

oh 

for  the  test  are  shown  in  Fig.  1.  A'  plug  commutator  p may 
be  used  to  make  connection  with  the  insulated  line  wire  L or 


272 


ELECTRICITY. 


with  one  side  of  the  condenser  c,  by  putting  a plug  in  1 or  2. 

On  depressing  the  key  k, 
contact  is  made  with  one 
pole  of  the  battery  B, 
having  about  100  cells; 
on  releasing  the  key,  the 
discharge  from  the  line 
or  the  condenser  passes 
through  the  galvanom- 
eter to  the  ground  at  O. 

Example.— T he  d e- 
fiection  through  a con- 
denser of  1.5  microfarads 
(mfds.)  was  82  divisions, 
and  through  a cable,  154  divisions.  Find  the  capacity  of  cable. 
Solution.— From  the  formula  given, 

154 

k2  = 1.5  X “oo"  ==  2-8  microfarads. 

Voltmeter  Method  of  Testing  Insulation.— An  ordinary  Weston 
voltmeter  with  a range  of  150  volts  has  a resistance  of  about 

19.000  ohms.  If,  then,  this  instrument  is  connected  across  a 
110-volt  circuit,  it  will  indicate  the  resistance  of  the  circuit, 
that  is,  of  itself,  since  the  resistance  of  the  armature  and 
leads  is  very  low.  If  v is  the  voltage  across  the  mains,  r the 
resistance  of  the  voltmeter,  and  x the  voltmeter  reading, 

V T 

then  the  resistance  to  be  determined,  R = — . When  the 

x 

voltmeter  is  put  across  the  mains,  v = 110,  r = 19,000,  and 
x = 110.  The  only  resistance  in  the  circuit  is  the  voltmeter 
11 0 V 1 9 OftO 

itself,  for  R = — - = 19,000  ohms.  If  we  now  put  in 

series  with  the  voltmeter  a high  resistance,  thereby  reducing 

110  X 19  000 

the  reading  to  2 divisions,  the  total  resistance  R = 2 ^ 

= 1,045,000  ohms.  From  this  we  must  subtract  the  voltmeter 
resistance  in  order  to  find  the  added  resistance,  which  is 

1.045.000  — 19,000  = 1,026,000  ohms.  A deflection  of  one 
division  gives  2,071,000  ohms.  To  obtain  higher  readings,  a 
special  high-resistance  voltmeter  should  be  used.  The  con- 
nections are  made  as  shown  in  Fig.  2,  where  V is  the 


CABLE  TESTING. 


273 


voltmeter,  F the  feeder,  and  D the  source  of  current.  If  I is  the 
insulation  resist- 
ance of  a feeder, 
the  corrected  for- 
mula becomes 

..  v r 
I = r. 


r 


Fig.  2. 


When  a voltmeter  is  used  having  a resistance  of  1 megohm 
(1,000,000  ohms),  then  a deflection  of  1 division,  when  con- 
nected up  as  shown,  would  give  an  insulation  resistance 


7 = 


110  x 1,000,000 


— 1,000,000  = 109  megohms. 


Loss-of-Charge  Method  of  Cable  Testing.— The  core  of  the 
cable  must  first  be  put  to  earth  a sufficient  length  of  time 
to  be  thoroughly  clear  from  any  charge  due  to  previous  elec- 
trification; then  the  far  end  is  freed,  and  connections  are 
made  as  shown  in  Fig.  3. 
On  depressing  the  key  k, 
the  cable  is  put  to  earth 
through  the  condenser  c, 
which  should  be  of  very  small 
capacity,  say  one-fiftieth  of  a 
microfarad.  Both  the  cable  L 
and  the  condenser  c are  then 
charged  from  the  battery  B by 
depressing  the  key  k',  and  on 
releasing  k,  the  condenser  is 
discharged  through  the  bal- 
listic galvanometer  g , a mo- 
ment being  chosen  when  the 
galvanometer  is  at  zero,  show- 
ing that  the  charge  is  steady.  The  deflection  produced  (di) 
represents  the  full  charge  held  by  the  cable.  The  key  k is 
then  again  depressed,  and  cable  and  condenser  are  charged 
for,  say,  half  a minute,  after  which  the  battery  is  discon- 
nected at  k',  and  leakage  of  the  charge  is  allowed  to  take 
place  for  perhaps  5 minutes.  Selecting  a moment  when  the 
charge  is  steady,  indicating  an  even  distribution,  the  key  k 
is  raised,  and  the  condenser  discharged  through  the 


274 


ELECTRICITY. 


galvanometer.  The  deflection  (d2)  obtained  will  be  less  than 
the  first  one,  owing  to  the  leakage  of  charge  during  the 
5 minutes,  and  will  therefore  be  a measure  of  the  conducting 
power  of  the  cable  covering,  or  its  insulation  resistance. 
The  ratio  of  these  two  deflections,  di  and  d2 , will  ordinarily 
be  sufficient  to  indicate  the  condition  of  the  cable  without 
further  calculation;  the  exact  insulation  resistance  may  be 
found  by  the  following  formula, 
x _ 26.06 1 

~ Klos% 

where  I = insulation  resistance  of  the  cable  in  megohms; 

t = time  in  minutes  during  which  the  charge  is 
allowed  to  leak; 

K = capacity  of  the  cable  in  microfarads; 
di  = initial  discharge  deflection; 
d2  — final  deflection  after  t minutes. 

Example. — In  a loss-of-charge  insulation  test,  the  initial 
deflection  was  238  divisions,  and  the  deflection  after  5 min- 
utes’ leakage  was  137  divisions.  The  capacity  of  the  cable 
being  1.8  microfarads,  what  was  the  insulation  resistance? 

Solution.—  I 26‘°— b = 301.8  megohms. 

1.8  X log  ~ 

The  battery  used  in  this  test  may  be  about  100  chloride-of- 
silver  cells,  or  the  same  number  of  Leclanch6  cells.  In  the 
latter  case  it  will  be  better  to  make  the  electrolyte  of  only 
about  one-fifth  the  usual  strength,  to  prevent  creeping  of  the 
salts,  as  only  very  small  currents  are  required  for  these  tests. 
The  battery  must  be  very  thoroughly  insulated. 

Location  of  Faults. 
A fault  in  a cable 
usually  develops 
slowly;  and  there  is 
\ considerable  resist- 
ance at  that  point; 
therefore,  in  determining  the  location  of  the  fault,  its  resist- 
ance must  be  taken  into  account.  Let  A B,  Fig.  4,  be  the 
cable,  and  let  a fault  F connect  to  the  ground  at  G through 


CABLE  TESTING. 


275 


a resistance  R.  When  the  end  B of  the  cable  is  insulated,  the 
resistance  >is  measured  at  the  station  A,  and  is  equal  to  the 
resistance  of  that  portion  of  the  cable  between  the  station 
and  the  fault  plus  the  resistance  of  the  fault,  that  is,  x + R. 
B is  then  grounded  at  G',  and  the  resistance  is 

+ y + R' 
x-\-  E = r. 
yr 


Let 

Let 

Let 

Then, 


x + y = r". 

x = r'  — j/  (r  — r')(r"  — r'); 
y = r"  — r'  + |/ (r  --  rf){r"  — r'). 

If  L = length  of  cable  in  feet,  the  distance  from  A to  the 
fault  is 

Lx 

z~-fy‘ 


Example. — The  resistance  of  a cable  in  good  condition  is 
3 ohms.  A fault  develops,  and,  on  testing,  the  resistance 
through  it  is  160  ohms,  the  far  end  of  the  cable  being  insu- 
lated. When  the  far  end  is  grounded,  the  resistance  is 
2.95  ohms.  What  is  the  distance  to  the  fault,  the  length  of 
cable  being  5,180  ft.? 

Solution.—  r = 160,  r*  =±=  2.95,  r"  = 3. 


Then,  x = 2.95  - 1/ 157.05  X .05  - .15  ohm. 

y ■=  3 — 2.95  + j/ 157.05  X -05  = 2.85  ohms. 
The  distance  to  the  fault  — 5.180  X -15  — 259  ft. 


3 


276 


SURVEYING. 


SURVEYING. 


COMPASS  SURVEYING. 

The  magnetic  bearing  of  a line  is  the  angle  that  the  line 
makes  with  the  magnetic  needle.  The  length  of  a line, 
together  with  its  bearing,  is  termed  a course.  To  take  the 
bearing  of  a line,  set  the  compass  directly  over  a point  in  it, 
at  one  extremity,  if  possible.  This  may  be  done  by  means  of 
a plumb-bob  suspended  from  the  compass. 

Bring  the  compass  to  a perfectly  level  position.  Let  a flag- 
man hold  a rod  carefully  plumbed  at  another  point  of  the 
line,  preferably  the  other  extremity,  if  he  can  be  distinctly 
seen.  Direct  the  sights  upon  this  rod  and  as  near  the  bottom 
of  it  as  possible.  Always  keep  the  same  end  of  the  compass 
ahead— the  north  end  is  preferable,  as  it  is  readily  distin- 
guished by  some  conspicuous  mark,  usually  a fleur-de-lis— and. 
always  read  the  same  end  of  the  needle,  that  is,  the  north 
end  of  the  needle  if  the  north  point  of  the  compass  is  ahead, 
and  vice  versa.  Before  reading  the  angle,  see  that  the  eye  is 
in  the  direct  line  of  the  needle,  so  ac  to  avoid  the  error  that 
would  otherwise  result  from  parallax , or  apparent  change  of 
the  position  of  the  needle,  due  to  looking  at  it  obliquely. 

The  angle  is  read  and  recorded  by  noting,  first , whether 
the  N or  S point  of  the  compass  is  nearest  the  end  of  the 
needle  being  read;  second , the  number  of  degrees  to  which  it 
points;  and  third , the  letter  E or  W nearest  the  end  of  the 
needle  being  read. 

Let  A B in  Fig.  1 be  the  direction  of  the  magnetic  needle, 
B being  at  the  north  end.  Let  the  sights  of  the  compass  be 
directed  along  the  line  CD.  The  north  point  of  the  compass 
will  be  seen  to  be  nearest  the  north  end  of  the  needle  which 
is  to  be  read.  The  needle,  which  has  remained  stationary 
while  the  sights  were  being  turned  to  CD,  now  points  to  45° 
between  the  N and  E points,  and  the  angle  is  read  north  forty- 
five  degrees  east  (N  45°  E). 

A sure  test  of  the  accuracy  of  a bearing  is  to  set  up  the 
compass  at  the  other  end  of  the  line,  i.  e.,  the  end  first  sighted 


COMPASS  SURVEYING. 


277 


Fig.  1. 


to,  and  sight  to  a rod  set  up  at  the  starting  point.  This  proc- 
ess  is  called  backsighting.  If  the  second 
bearing  is  the  same  as  the  first,  the  reading 
is  correct.  If  it  is  not  the  same,  it  shows 
that  there  is  some  disturbing  influence  at 
either  one  or  the  other  end  of  the  line.  To 
determine  which  of  these  two  bearings  is  the 
true  one,  the  compass  must  be  set  up  at 
one  or  more  intermediate  points,  when 
two  or  more  similar  bearings  will  prove 
the  true  one. 

The  magnetic  meridian  is  the  direction  of  the  magnetic 
needle.  The  true  meridian  is  a true  north 

tn,4T¥ro  and  south  line,  which,  if  produced,  would 
| jj  L pass  through  the  poles  of  the  earth.  The 

0 declination  of  the  needle  is  the  angle  that 

the  magnetic  meridian  and  the  true 
meridian  make  with  each  other. 

Example  of  the  Use  of  the  Compass  in  Rail' 
road  Work.— Suppose  CAD  in  Fig.  2 to  be 
a railroad  in  operation,  and  that  it  has 
been  decided  to  run  a compass  line  from 
the  point  A along  the  valley  of  the  stream 
X to  the  point  B.  The  bearing  of  the 
tangent  A D cannot  be  determined  by  set- 
ting up  the  compass  at  A on  account  of  the 
attraction  of  the  rails.  The  direction  of 
this  tangent,  however,  can  be  obtained  by 
setting  up  at  A and  sighting  to  a flag  held 
at  D.  The  point  A,  which  is  the  starting 
point  of  the  line  to  be  run,  is  marked  0. 
Producing  the  line  A D 440  ft.,  the  point  E 
is  reached,  which  has 
been  previously  de- 
cided on  as  a proper 
place  for  changing  the 
r>  direction  of  the  line. 
The  compass  having 


Fig.  2. 


been  set  up  at  E , the  bearing  of  the  line  A E , which  is  the 


278 


SURVEYING 


line  A D produced,  is  found  by  sighting  to  A,  or,  what  is  still 
better,  to  the  point  D,  if  that  point  can  be  seen.  The  number 
of  Sta.  (Station)  E , namely,  4 + 40,  and  the  bearing  of  A E are 
then  recorded  by  the  compassman.  By  this  time  the  chief  of 
party  has  located  the  point  Fy  and  the  flag  is  in  place  for 
sighting.  The  axmen,  if  there  is  work  for  them  to  do,  are 
now  put  in  line  by  the  head  chainman;  the  axmen  clear 
only  so  much  as  would  interfere  with  rapid  chaining.  The 
bearing  of  the  line  EF  haying  been  recorded,  the  compass  is 
moved  quickly  to  F,  replacing  the  target  left  by  the  flagman, 
leveled  up,  and  directed  toward  the  point  G , which  is  already 
located.  The  chainmen  reaching  F,  its  number  11  + 20  is 
recorded  by  the  compassman  and  the  instrument  sighted  to  G 
and  the  work  continued  as  before. 


FORM  FOR  KEEPING  NOTES. 

A plain  and  convenient  form  for  keeping  compass  notes  is 
the  form  given  on  page  279,  which  is  a record  of  the  survey 
platted  in  Fig.  2.  The  first  column  of  the  table  contains  the 
station  numbers,  the  notation  running  from  the  bottom  to  the 
top  of  the  page.  By  means  of  this  arrangement,  the  lengths 
of  the  courses  are  found  by  subtracting  the  number  of  the 
station  of  one  compass  point  from  the  number  of  the  station 
of  the  next  succeeding  compass  point.  Before  work  has 
commenced  on  the  plat,  the  subtractions  are  made  and  the 
lengths  of  the  courses  are  written  in  red  ink  between  the 
station  numbers. 

The  second  column  contains  the  bearings  of  the  lines. 
The  bearing  recorded  opposite  to  a station  is  the  bearing  at 
the  course  between  the  given  station  and  the  one  next  above. 
Thus,  the  bearing  recorded  opposite  Sta.  0 is  75°  00'  W,  and  is 
the  bearing  of  the  line  extending  from  Sta.  0 to  Sta.  4 + 40 
next  above.  The  length  of  the  course  is  the  difference 
between  0 and  4 + 40  equal  to  440  ft.  The  bearing  recorded 
opposite  to  4 + 40  is  N 25°  00'  W.  It  is  the  bearing  of  the  line 
extending  from  Sta.  4 + 40  to  Sta.  11  + 20  next  above.  Its 
length  is  found  by  subtracting  4 + 40  from  11  + 20  equal  to 
680  ft.,  and  so  on. 


TRANSIT  SURVEYING. 


279 


In  the  third  column,  under  the  head  of  remarks,  are 
recorded  notes  of  reference,  topography,  and  any  informa- 
tion that  may  aid  in  platting  or  subsequent  location. 


Station, 

Bearing. 

Remarks. 

47  + 75 

End  of  line. 

35  + 75 

N 25°  40'  E 

27  + 50 

N 14°  10'  E 

20  + 35 

N 2°  30'  W 

Woodland. 

11  + 20 

N 15°  10'  W 

4 + 40 

N 25°  00'  W 

0 

N 75°  00'  W 

Sta.  0 is  at  P.  C,  of  14°  curvejo 
left  at  Bellford  Sta.  O.  & P.  R.  R. 

TRANSIT  SURVEYING. 

The  Vernier. — A vernier  is  a contrivance  for  measuring 
smaller  portions  of  space  than  those  into  which  a line  is 
actually  divided.  The  divided  circle  of  the  transit  is  gradu- 
ated to  half  degrees,  or  30'.  The  graduations  on  the  verniers 
run  in  both  directions  from  its  zero  mark,  making  two  dis- 
tinct verniers,  one  for  reading  angles  turned  to  the  right  and 
the  other  for  reading 
those  turned  to  the  left. 

In  reading  the  vernier, 
the  observer  should 
first  note  in  which 
direction  the  gradu-  pIG  ^ 

ations  of  the  divided 

circle  run.  In  Fig.  1 the  graduations  increase  from  left  to 
right  and  extend  from  57°  to  91°.  Next,  he  should  note  the 
point  where  the  zero  mark  of  the  vernier  comes  on  the 
divided  circle.  In  Fig.  1 the  zero  mark  cOmes  between  74° 
and  74£°.  Now,  as  the  circle  graduations  read  from  left  to 
right,  we  read  the  right-hand  vernier  and  find  that  the  23d 
graduation  on  the  vernier  coincides  with  a graduation  on  the 


280 


SURVEYING. 


divided  circle  and  the  vernier  reads  23',  which  we  add  to  74°, 
making  a reading  of  74°  23',  an  angle  to  the  left.  In  Fig.  2 the 
graduations  on  the  circle 
increase  from  right  to 
left,  and  we  accordingly- 
read  the  left-hand  ver- 
nier. The  zero  mark  of 
the  vernier  comes  be- 
tween  67£°  and  68°. 

Reading  the  vernier,  we 
find  that  the  13th  graduation  on  the  vernier  coincides 
with  a graduation  on  the  circle  and  the  vernier  reads  13'. 
Accordingly,  we  add  to  67£°,  the  reading  = 13',  making  a 
total  reading  of  67°  43',  an  angle  to  the  right. 

Setting  Up  the  Instrument.— In  setting  up  a transit,  three 
preliminary  conditions  should  be  met  as  nearly  as  possible: 

1.  The  tripod  feet  should  be  firmly  planted. 

2.  The  plate  on  which  the  leveling  screws  rest  should  be 
level. 


Fig.  2. 


3.  The  plumb-bob  should  be  directly  over  the  given  point. 
When  these  three  conditions  are  met,  the  completion  of 
the  operation  is  quickly  performed  with  the  leveling  screws. 

How  to  Prolong  a Straight  Line.— Let  A B,  in  Fig.  3,  be  a 
straight  line  which  it  is  required  to  prolong  or  “produce.,, 


Fig.  3. 


400' 


The  line  can  be  prolonged  in  two  ways:  by  means  of 
foresight  or  by  means  of  backsight. 

1.  By  foresight,  set  up  the  transit  at  A and  sight  to  B; 
measure  400  ft.  from  B in  the  opposite  direction  from  A. 
Then,  by  means  of  signals,  move  the  flag  to  the  right  or  left 
until  the  vertical  cross-hair  shall  exactly  bisect  the  flag  held 
at  C.  Then,  the  line  B C will  be  the  prolongation  of  the 
line  AB. 

2.  By  backsight,  set  the  transit  at  B and  sight  to  A. 
Reverse  the  telescope,  and  having  measured  400  ft.  from  B in 
the  opposite  direction  from  A,  set  the  flag  at  C ; then  will  the 
line  B C be  the  line  A B produced. 


TRANSIT  SURVEYING. 


281 


Horizontal  Angles  and  Their  Measurement.— A horizontal  angle 
is  one  the  boundary  lines  of  which  lie  in  the  same  horizontal 
plane.  Let  A,  B , and  C , in  Fig.  4,  be  three  points,  and  let  it 
be  required  to  find  the  horizontal  angle  formed  by  the  lines 
A B and  AC  joining  these  # 0 

points.  Set  up  the  instrument  v*3°3<> 

precisely  over  the  point  A,  and  / 

carefully  level  it.  Set  the  ver-  _ / 

nier  at  zero,  and  place  flags  at  B 
B and  C.  Sight  to  the  flag  at  FlG-  4* 

B and  set  the  lower  clamp.  Then,  by  means  of  the  lower 
tangent  screw,  cause  the  vertical  cross-hair  to  exactly  bisect 
the  flag  at  B.  Loosen  the  upper  clamp.  With  a hand  on 
& either  standard,  turn  the  telescope  in  the  same 

direction  as  that  of  the  hands  of  a watch  until 
the  flag  at  C is  covered  or  nearly  covered  by 
the  vertical  cross-hair.  Clamp  the  upper 
plate,  and  with  the  upper  tangent  screw  bring 
the  line  of  sight  exactly  on  the  flag  at  C.  The 
arc  of  the  graduated  circle  traversed  by  the 
zero  point  of  the  vernier  will  be  the  measure 
of  the  angle  BAC , as  143°  30'.  The  points  A, 
B,  and  Care  not  necessarily  in  the  same  hori- 
zontal plane,  but  the  level  plate  of  the  instru- 
ment projects  them  into  the  horizontal  plane 
.©  in  which  it  revolves. 

A Deflected  Line.— A deflected  line,  or  “ angle 
line,”  is  a consecutive  series  of  lines  and 
angles.  The  direction  of  each  line  is  referred 
to  the  line  immediately  preceding  it,  the  latter 
being,  in  imagination,  produced,  and  the 
angle  measured  between  it  and  the  next  line 
actually  run.  The  angles  are  recorded  RT  or 
LT,  according  as  they  are  turned  to  the  right 
or  left  of  the  prolongation  of  the  immediately 
preceding  line.  An  example  of  a deflected 
line  is  shown  in  Fig.  5;  it  starts  from  the  head 
block  of  switch  at  Benton  Station, O.  & P.  R.  R. 

Set  up  the  transit  at  A with  vernier  at  zero.  Sight  to  a flag 


282 


SURVEYING. 


held  at  F on  the  center  line  of  the  track,  O.  & P.  R.  R. 
Loosen  the  vernier  clamp,  the  point  B being  determined,  and 
turn  the  telescope  until  the  point  B is  distinctly  seen;  clamp 
the  vernier,  and  accurately  sight  to  flag  held  at  B\  the  angle 
reads  32°  30'  and  is  recorded  RT  32°  30',  with  a sketch  showing 
the  connection.  The  bearing  of  the  line  A B cannot  be  taken 
at  A on  account  of  the  attraction  of  the  rails.  The  point  A is 
in  the  head  block  of  the  switch  (which  is  designated  by  the 
abbreviation  H.  B.)  at  Benton  Station,  O.  & P.  R.  R.  The 
instrument  is  now  moved  to  B , the  vernier  set  at  zero  and 
backsighted  to  A;  the  bearing  of  A B , viz.,  N 75°  00/  E,  is 
taken,  and  the  number  of  station  B , viz.,  2 + 90,  together 
with  the  bearing  of  A B recorded.  The  telescope  is  then 
reversed,  pointing  in  the  direction  BB'.  The  point  C being 
determined,  the  upper  clamp  is  loosened  and  the  telescope 
turned  to  the  right  and  sighted  to  C.  The.  reading  is  found 
to  be  14°  30'  and  recorded  RT  14°  30'.  It  measures  the  angle 
B'  B C.  The  bearing  N 89°  20'  E is  then  recorded.  The 
instrument  is  next  set  up  at  C,  the  vernier  set  at  zero,  back- 
sighted  to  B,  and  then  reversed;  the  deflection  to  D,  viz., 
RT  10°  00'  read  and  recorded,  together  with  the  number  of  the 
station  at  (7,  viz.,  6 + 85.  This  deflection  measures  the  angle 
C' CD  and  gives  the  direction  of  the  line  CD.  A good  form 
of  notes  for  such  a survey  is  the  following: 


Station 

Deflection. 

Mag.  Bearing. 

Ded.  Bearing. 

Remarks. 

13+63 

End  of  Lin 

ie. 

10+31 

LT30°00f 

N.  69°Z5’E. 

E.  69°30'E. 

6+85 

Rr10W 

S.  80°30'E. 

S.  80°30'E- 

9+90 

R*14°3<y 

S.  89°20’ E. 

jr.89°30'E. 

% 

TT  Ti  nf  fhnitfik 

0 

N.  75°0QfE. 

Sta.o\ 

at  Benton  Sta. 

Checking  Angles  by  the  Needle.— In  spite  of  the  greatest  care, 

errors  in  the  reading  and  recording  of  angles  will  occur- 
The  best  check  to  such  errors  is  the  magnetic  needle. 

In  Fig.  6,  we  have  an  example  of  the  use  of  the  needle  in 
checking  angles.  The  bearing  of  the  line  A B,  which  corre- 
sponds to  A B in  Fig.  5,  is  N 75°  00'  E,  and  is  assumed  to  be 
correct.  The  bearing  of  the  line  B C , as  read  from  the  needle, 


TRIANGULATION. 


283 


is  N 89°  20'  E.  Its  deduced  bearing  is  obtained  as  follows:  To 
the  bearing  of  the  line  A B,  viz.,  N 75°  00'  E,  we  add  the  RT 
deflection  14°  30';  the  sum  is  89°  30',  which  is  recorded  in  the 
column  headed  Ded.  Bearing.  The  deduced  bearing,  it  will 
be  seen,  is  10  minutes 
greater  than  the  mag- 
netic bearing  read  from 
the  needle.  Had  the 
deflection  angle  been 
recorded  LT  instead  of 
RT,  the  deduced  bear- 
ing would  have  been 
the  difference  between 
75°  00'  and  14°  30',  which 
is  60°  30',  and  would  be  pIG>  6. 

recorded  N 60°  30'  E. 

The  magnetic  bearing  being  N 89°  20'  E,  would  have  at  once 
revealed  the  error.  The  confusion  of  the  directions  RT  and 
LT  is  the  commonest  source  of  error  in  recording  deflections, 
though  sometimes  a mistake  of  10  degrees  is  made  in  reading 
the  vernier.  Both  angle  and  bearing  should  be  read  after 
they  are  recorded,  and  compared  with  the  recorded  readings. 


TRIANGULATION. 

Triangulation  is  an  application  of  the  principles  of  trigo- 
nometry to  the  calculation  of  inaccessible  lines  and  angles. 

A common  occasion  for  its  use  is 
illustrated  in  Fig.  1,  where  the  line 
of  survey  crosses  a stream  too  wide 
and  deep  for  actual  measurement. 
Set  two  points  A and  B on  line, 
one  on  each  side  of  the  stream. 
Estimate  roughly  the  distance  A B. 
Suppose  the  estimate  is  425  ft.  Set 
another  point  C,  making  the  dis- 
tance A C equal  to  the  estimated 
Set  the  transit  at  A and  measure  the 
Next  set  up  at  the  point  C and 


distance  A B = 425  ft. 
angle  BA  C = say,  79° 00'. 


284 


SURVEYING. 


measure  the  angle  A CB  = say,  56°  20'.  The  angle  ABC  is 
then  determined  by  subtracting  the  sum  of  the  angles  A and 
C from  180°;  thus,  79°  00' + 56°  20'  = 135°  20';  180°  00' - 135°  20' 
_ 440  4(y  = the  angle  ABC.  We  now  have  a side  and  three 
angles  of  a triangle  given,  to  find  the  other  two  sides  A B and 
CB.  In  trigonometry,  it  is  demonstrated  that,  in  any  triangle 
the  sines  of  the  angles  are  proportional  to  the  lengths  of  the  sides 
opposite  to  them.  In  other  words,  sin  A : sin  B = B C:  AC;  or, 
sin  A : sin  C = B C : A B,  and  sin  B : sin  C = A C : A B. 

Hence,  we  have  sin  44°  40' : sin  56°  20'  = 425  : side  A B\ 
sin56°20/  = .83228; 

.83228  X 425  = 353.719; 
sin  44°  40'  = .70298; 

353.719  -r-  .70298  = 503.17  ft.  = side  A B. 

Adding  this  distance  to  76  + 15,  the  station  of  the  point  A , 
we  have  81  + 18.17,  the  station  at  B. 

Another  case  is  the  following:  Two  tangents,  A B and  C D 
(see  Fig.  2),  which  are  to  be  united  by  a curve,  meet  at  some 
inaccessible  point  E.  Tangents  are  the  straight  portions  of  a 


A B,  and  two  points  Uand  D of  the  tangent  CD , being  care- 
fully located,  set  the  transit  at  B,  and  backsighting  to  A , 
measure  the  angle  EB  C = 21°  45';  set  up  at  C,  and,  back- 
sighting  to  D,  measure  the  angle  ECB  = 21°  25'.  Measure 
the  side  B C — 304.2  ft. 

Angle  C EF  being  an  exterior  angle  of  triangle  EB  C equals 
sum  of  EB  C and  ECB  — 21°  45'  + 21°  25'  = 43°  10';  angle  B E C 
— 180°  — CEF  — 136°  50'.  From  trigonometry,  we  have 


sin  136°  50' : sin  21°  45'  = 304.2  ft.  : CE) 
sin  21°  45'  = .37056; 

.37056  X 304.2  = 112.724352; 
sin  136°  50'  = .68412; 

side  CE  « 112.724352  -4-  .68412  = 164.77  ft. 


Fig.  2. 


r 


line  of  railroad.  The 
angle  CEF,  which  the 
tangents  make  with 
each  other,  and  the  dis- 
tances BE  and  CE  are 
required.  Two  points 
A and  B of  the  tangent 


TRI  ANGULATION. 


285 


Again,  we  find  B E by  the  following  proportion: 
sin  136°  50' : sin  21°  25'  = 304.2  : side  B E; 
sin  21°  25'  = .36515; 


-J50L- 


-f- 


i~ 


/ 


.36515  X 304.2  = 111.07863; 
sin  136°  50'  = .68412; 

side  B E = 111.07863  .68412  = 162.36  ft. 

A building  H,  Fig.  3,  lies  directly 'in  the  path  of  the  line 
A Bf  which  must  be  produced  beyond  H.  Set  a plug  at  B, 
and  then  turn  an  angle  BBC 

= 60°.  Set  a plug  at  C in  the  4 „ 

line  B C , at  a suitable  distance  \ 

from  B,  say,  150  ft.  Set  up  at  C, 
and  turn  an  angle  B CD  = 60°, 
and  set  a plug  at  D,  150  ft.  from  \ / 

C.  The  point  D will  be  in  the  # 

prolongation  of  A B.  Then,  set  Fig.  3. 

up  at  Dy  and  backsighting  to 

Cy  turn  the  angle  C D D'  — 120°.  D D'  will  be  the  line 
required,  and  the  distance  B D 
will  be  150  ft.,  since  BCD  is 
an  equilateral  triangle. 

A B and  CD,  Fig.  4,  are  tan- 
gents intersecting  at  some  in- 
accessible point  H.  The  line 
A B crosses  a dock  O P,  too 
wide  for  direct  measurement, 
and  the  wharf  LM.  F is  a 
point  on  the  line  A B at  the 
wharf  crossing.  It  is  required 
to  find  the  distance  B H and 
the  angle  FHG.  At  B , an 
angle  of  103°  30'  is  turned  to  the 
left  and  the  point  E set  217' 
from  B = to  the  estimated  dis- 
tance BF.  Setting  up  at  E , 
the  angle  B E F is  found  to 
be  39°  00'. 

Whence,  we  find  the  angle 
(103°  30'  + 39°)  = 37°  30', 


BFE  — 180°  - 


286 


SURVEYING. 


From  trigonometry,  we  have 

sin  37°  30' : sin  39°  00'  = 217  ft. : side  BF\ 
sin  39°  00'  = .62932; 

.62932  X 217  = 136.56244; 
sin  37°  30'  = .60876; 

side  B F = 136.56244  -f-  .60876  = 224.33  ft. 

Whence,  we  find  station  F to  be  20  4-  17  4-  224.33  = 22 
+ 41.33.  Set  up  at  F and  turn  an  angle  HFG  = 71° 00'  and 
set  up  at  a point  G where  the  line  CD  prolonged  intersects 
FGo  Measure  the  angle  FGH  = 57° 50',  and  the  side  FG 
= 180.3.  The  angle  FHG  = 180°  - (71°  + 57°  50')  = 51°  10'. 
From  trigonometry  we  have 

sin  51°  10'  : sin  57°  50'  = 180.3  : side  FH. 

Sin  57°  50'  = .84650;  .84650  X 180.3  = 152.62395;  sin  51°  W 
= .77897;  side  FH  = 152.62395  -f-  .77897  = 195.93  ft.;  whence 
we  find  station  H to  be  24  4-  37.26. 


CURVES. 

Two  lines  forming  an  angle  of  1°  with  each  other  will,  at 
a distance  of  100  ft.  from  the  angular  point,  diverge  by  1.745  ft. 

The  degree  of  a curve  is  deter- 
mined by  that  central  angle 
which  is  subtended  by  a chord 
of  100  ft.  Thus,  if  BOG 
(Fig.  1)  is  10°  and  B G is  100  ft., 
BGHKC is  a 10°  curve. 

The  deflection  angle  of  a 
curve  is  the  angle  formed  at 
any  point  of  the  curve  between 
a tangent  and  a chord  of  100  ft. 
The  deflection  angle  is  there- 
fore half  the  degree  of  the  curve. 
Thus,  if  the  chord  B G is  100  ft., 
the  angle  EBG  is  the  deflec- 
tion angle  of  curve  BGHKC, 
and  is  half  the  angle  BOG. 
Example— Given,  the  deflection  angle  EBG  = D (Fig.  1), 
to  find  the  radius  B 0 = R. 


CURVES. 


287' 


Solution.— Draw  0 L perpendicular  to  B G.  In  the  right- 

JB  L 

angled  triangle  BOL,  we  have  sin  BOL  = but  BOL 

= E B G — D,  since  0 L , being  perpendicular  to  the  chord 
B G,  bisects  the  arc  B LG.  But  the  angle  D = \ B 0 G;  hence, 
angle  BOL  = D.  BL  = 50  ft.,  and  the  radius  B O = R. 
Substituting  these  values  in  the  given  equation,  we  have 

sin  1)  — 4?;  whence,  R sin  D — 50,  and  R = — — — . 

R an  L 


For  curves  of  from  1°  to  10°,  the  radius  may  be  found  by 
dividing  5,780  ft.  (the  radius  of  a 1°  curve)  by  the  degree  of 
the  curve.  The  results  obtained  are  sufficiently  accurate  for 
all  practical  purposes.  For  sharp  curves,  i.  e.,  for  those 

50 

exceeding  10°,  the  above  formula,  viz.,  R = — — should  be 

sin  D 

used,  especially  if  the  radius  is  to  be  used  as  a basis  for 
further  calculation. 

Tangent  Distances.— When  an  intersection  of  tangents  has 
been  made  and  the  intersection  angle  measured,  the  next 
question  is  the  degree  of  curve  that  is  to  unite  them,  which 
being  decided,  the  next  step  in  order  is  the  location  of  the 
points  on  the  tangents  where  the  curve  begins  and  ends. 
These  two  points  are  equally  distant  from  the  point  of 
intersection  of  the  tangents,  which  is  called  the  P.  I.  The 
point  where  the  curve  begins  is  called  the  point  of  curve , or 
the  P.  C.,  the  point  where  the  curve  terminates  is  called  th^ 
point  of  tangent , or  the  P.  T.  The  distance  of  the  P.  C.  and 
P.  T.  from  the  P.  I.  is  called  the  tangent  distance. 

In  Fig.  1,  let  A B and  CD  be  tangents  intersecting  at  the 
point  E and  forming  an  angle  CEF  = 40°  00'  with  each 
other.  It  is  decided  to  unite  these  tangents  by  a 10°  curve, 
whose  radius  is  573.7  ft.  Call  the  angle  of  intersection  J,  the 
radius  B O,  R , and  the  tangent  distance  B E,  T.  From  geom- 
etry we  know  that  B 0 C = CEF,  hence  the  angle  B O E 
= \CEF.  From  the  right  triangle  E B 0,  we  have  tan 

£0£=§l- 

Substituting  the  above  equivalents,  we  have  tan  1 1 = 

ot  T = R tan  *1;  R = 573.7;  * / = 20°;  tan  20°  = .36397; 
xC 


288 


SURVEYING. 


573.7  X .36397  = 208.81  ft.  Measure  back  from  the  point  E on 
both  tangents  the  distance  208.81  ft.  to  the  points  B and  C. 
Drive  plug  flush  with  the  ground  at  both  points  and  set 
accurate  center  points,  marked  by  tacks,  in  both.  Directly 
opposite  each  of  these  plugs  drive  a stake,  called  a guard 
stake  because  it  guards  or  rather  indicates  where  the  plug  is. 
The  stake  at  B,  if  the  numbering  of  the  stations  runs  from 
B toward  C,  will  be  marked  P.  C.,  and  the  stake  at  C will  be 
marked  P.  T. 

To  Lay  Out  a Curve  With  a Transit. — Having  set  the  tangent 
points  B and  C,  Fig.  1,  set  up  the  transit  at  B,  the  P.  C.  Set 
the  vernier  at  zero  and  sight  to  E , the  intersection  point. 
Suppose  B to  be  an  even  or  “ full  station,”  say  18,  and  that  it 
has  been  decided  to  set  stakes  at  each  hundred  feet.  Let  the 
central  angle  BOG , measured  by  the  100-ft.  chord  B G,  be 
10°;  then,  the  deflection  angle  E B G,  whose  vertex  B is  in 
the  circumference  and  subtended  by  the  same  chord  B G, 
will  be  £ B 0 G,  or  5°.  Turn  an  angle  of  5°  from  B,  which  in 
this  case  will  be  to  the  right,  measure  a full  chain  100  ft. 
from  B and  line  in  the  flag  at  G;  drive  a stake  at  G,  which 
will  be  marked  19.  Turn  off  an  additional  5°  making  10° 
from  zero,  and  at  the  end  of  another  chain  from  G,  at  H,  set 
at  a stake  marked  20.  Continue  turning  deflections  of  5° 
until  20°  or  one-half  of  the  intersection  angle  is  reached. 
This  last  deflection,  if  the  work  has  been  correctly  done,  will 
bring  the  head  chainman  to  the  point  of  tangent  C.  It  is  but 
rarely  that  the  P.  C.  comes  at  a full  station.  When  the  P.  C. 
comes  between  full  stations  it  is  called  a substation , and  the 
chord  between  it  and  the  next  full  station  is  called  a sub- 
chord. Had  the  P.  C.  come  at  a substation,  say  17  + 32,  the 
deflection  for  the  subchord  of  100  — 32,  or  68  ft.,  the  distance 
to  the  next  station,  is  found  as  follows:  The  deflection  for  a 
full  station,  i.  e.,  100  ft.,  is 5°  = 300',  and  the  deflection  for  1 ft. 

is  ^5^.  = and  for  68  ft.  the  deflection  will  be  68  X 3 = 204' 

= 3°  24',  which  is  turned  off  from  zero  and  a stake  set  on  line, 
68  ft.  from  the  transit,  at  station  18.  The  length  of  a curve 
uniting  two  given  tangents  whose  intersection  is  determined, 
is  found  as  follows: 


CURVES. 


289 


* 

Suppose  I = 32°  40'  and  that  the  tangents  are  to  be  united 
by  a 6°  curve.  32°  40'  reduced  to  the  decimal  form  is  32.667°; 
as  each  central  angle  of  6°  will  subtend  a 100-ft.  chord  or  one 
chain,  there  will  be  as  many  such  chords  or  chains  as  the 
number  of  times  6 is  contained  in  32.667,  which  is  5.444,  that 
is,  there  will  be  5.444  chains  in  the  curve,  or  544.4  ft.,  which  is 
the  required  length  of  the  curve.  The  P.  C.  and  P.  T.  having 
been  set  and  the  station  of  the  P.  C.  determined  by  actual 
measurement,  say  58  + 71,  the  station  number  of  the  P.  T.  is 
found  by  adding  to  58  + 71,  the  station  number  of  the  P.  C., 
the  calculated  length  of  the  curve  544.4  ft.  58  + 71  -f  544.4  = 
64  + 15.4,  the  station  of  the  P.  T. 

Tangent  and  Chord  Deflections. — Let  A B in  Fig.  2 be  a tan- 
gent, and  B C EH  a curve  commencing  at  B.  Produce  the 
tangent  A B to  the  point  JD.  The  line  CD  is  a tangent  deflec- 
tion, and  is  the  perpendicular  distance  from  the  tangent  to 
the  curve.  If  the  chord 
B C is  produced  to  the 
point  G , making  CG  = 

BC  = CE,  the  distance 
G E is  a chord  deflection 
and  is  double  the  tan- 
gent deflection  D C. 

Given,  the  radius 
BO  — R,  Fig.  2,  to  find 
the  chord  deflection 
EG  and  the  tangent 
deflection  CD  = FE. 

The  triangles  0 CE  and  CE  G are  similar,  since  both  are 
isosceles,  and  the  angle  G CE  = angle  COE.  Hence,  we  have 
0 C:  CE  = CE:  EG.  Denoting  the  chord  CE  by  c and  the 
chord  deflection  E G by  d,  we  have,  from  the  above  propor- 
c2 

tion,  R : c = c : d.  Therefore,  d — To  find  the  tangent 
R 

deflection,  draw  C F to  the  middle  point  of  EG.  Then  FE 
is  equal  to  the  tangent  deflection,  or  D C.  Hence,  the  tan- 
gent deflection  is  equal  to  one-half  the  chord  deflection,  or 
c2 

the  tangent  deflection  = — 
z n 


290 


SURVEYING 


If  the  P.  C.  does  not  fall  at  a full  station  (and  this  is 
usually  the  case),  compute  the  chord  deflection  by  substituting 
for  c in  the  formula  for  chord  deflection  £ c (c  + c').  Where 
c'  is  the  length  of  the  chord  from  the  P.  C.  to  the  full  station; 
or  if  the  tangent  deflection  / for  a chord  of  100  feet  has  been 
previously  found,  the  chord  deflection  for  the  second  station 

beyond  the  P.  C.  is  d0  =/  (l  -f  . 

Laying  Out  Curves  Without  a Transit.— During  construction, 

the  engineer  is  often  called  upon  to  restore  center  stakes  on 
a curve  when  the  transit  is  not  at  hand.  This  can  be  accom- 
plished reasonably  well  with  a tape,  as  follows: 

In  Fig.  3,  A B is  a tangent  and  P,  at  Sta.  8 -f  25,  is  the  P.  C. 
of  a 4°  curve;  a stake  is  required  at  each  full  station.  The 
stakes  at  A and  B are  restored,  determining  the  P.  C.  and  the 
direction  of  the  tangent.  For  a 4°  curve  the  regular  chord 
deflection  for  100  feet  is  6.98  ft., 
and  the  tangent  deflection  is 
6.98  -4-  2 = 3.49  ft. 

The  distance  from  the  P.  C. 
to  the  next  station  C is  75  ft.; 
hence,  the  tangent  deflection 
C F = 752  -i-  (2  X 5,730  -4-  4)  = 
1 96  ft.  The  point  F is  found 
by  first  measuring  75  feet  from  B,  thus  locating  the  point  C)  in 
the  line  with  A B,  then  from  C measuring  C F = 1.96  feet, 
at  right  angles  to  B C\  the  point  F thus  determined  will  be 
Station  9.  Next,  the  chord  B F is  prolonged  100  feet  to  D;  as 
B F is  only  75  feet,  D G = d0  = 3.49  X (1  + t7t&)  = 6.11  feet. 
This  distance  is  measured  at  right  angles  to  B D;  the  point  G 
thus  determined  will  be  Station  10.  The  position  of  Station 
11,  the  P.  T.,  is  determined  in  the  same  manner,  except  that, 
as  the  chords  F G and  G H are  each  100  feet  long,  the 
regular  chord  deflection  of  6.98  feet  is  used  for  EH.  A stake 
is  driven  at  each  station  thus  located. 

To  Determine  Degree  of  Curve  by  Measuring  a Middle  Ordinate. 
In  track  work,  it  is  often  necessary  to  know  the  degree  of  a 
curve  when  no  transit  is  available  for  measuring  it.  The 
degree  can  be  found  by  measuring  the  middle  ordinate  of  any 


CURVES. 


291 


convenient  chord,  and  multiplying  its  length  by  8,  which 
will  give  the  chord  deflection  for  that  curve. 

Let  A B,  in  Fig.  4,  be  a 50-ft.  chord,  measured  on  the  track, 
and  let  the  middle  ordinate  ab  be  .44  ft.  .44  X 8 = 3.52 
= chord  deflection  for  50  ft.,  which,  expressed  in  decimal 
parts  of  a full  station,  is  ".5;  .52  = 

.25.  The  chord  deflection  for  100  ft.  ^ 

multiplied  by  .25  = the  chord  de- 
flection  for  50  ft.,  which  we  know 
by  calculation  to  be  3.52  ft.  Hence,  ^IG* 

3.52  h-. 25  = 14.08  ft.,  the  chord  deflection  for  100  ft.,  which, 
if  divided  by  1.745,  the  chord  deflection  for  a 1°  curve,  gives  a 
quotient  of  8.07,  nearly.  The  inference  is  that  the  curve  is  8°. 

How  to  Keep  Transit  Notes.— A good  form  for  location 
notes  is  the  following: 


DtflocUem. 

Tot- Angle. 

tfaff.  Bearing 

Dm*.  Bearing 

Bern 

June  JO  1894 

irks. 

9 

8 

1 

8+96 

4*54’ P.T. 

16*00 ‘ ~ 

S-  3*8*  B. 

2r.36°l*£. 

6+60 

4*0* 

6 

3*0* 

6+60 

3*0* 

6+80 

6 

IaOO' 

6+60 

6+60 

8*3* 

6*1 r 

4 

1*3* 

Ini.  Angle- 15*00' 

4*Cnrvt  BT 

3+60 

0*3* 

T- 188.81 ft. 

fief.  Angle  for  60  flrl'OO 

3+30 

P.C4*Br 

PC-  3+30 

Def. Angle  for  1 fl  - 1-8“ 

3 

Length  of  Ourve-376fi 

» 

PJP-C+96 

1 

O 

y.  zone's. 

B 80*1** 

In  the  first  column  the  station  numbers  are  recorded.  In 
the  second  column  are  recorded  the  deflections  with  the 
abbreviations  P.  C.  and  P.  T.,  together  with  the  degree  of 
curve  and  the  abbreviation  RT  or  LT,  according  as  the  line 
curves  to  the  right  or  left.  At  each  transit  point  on  the 
curve,  the  total  or  central  angle  from  the  P.  C.  to  that  point 
is  calculated  and  recorded  in  the  third  column.  This  total 
angle  is  double  the  deflection  angle  between  the  P.  C.  and 
the  transit  point.  In  the  above  notes  there  is  but  one  inter- 
mediate transit  point  between  the  P.  C.  and  P.  T.  The 


292 


SURVEYING. 


deflection  from  P.  C.  at  Sta.  3 + 20  to  the  intermediate  transit 
point  at  Sta.  4 -f  50  is  2°  36'.  The  total  angle  is  double  this 
deflection,  or  5°  12',  which  is  recorded  on  the  same  line  in 
the  third  column.  The  record  of  total  angles  at  once  indi- 
cates the  stations  at  which  transit  points  are  placed.  The 
total  angle  at  the  P.  T.  will  be  the  same  as  the  angle  Of  inter- 
section, if  the  work  is  correct.  When  the  curve  is  finished, 
the  transit  is  set  up  at  the  P.  T.,  and  the  bearing  at  the  for- 
ward tangent  taken,  which  affords  an  additional  check  upon 
the  previous  calculations.  The  magnetic  bearing  is  recorded 
in  the  fourth  column,  and  the  deduced  or  calculated  bearing 
is  recorded  in  the  fifth  column. 


LEVELING. 

Examples  in  Direct  Leveling.— The  principles  of  direct  level- 
ing are  illustrated  in  the  figure. 

Let  A be  the  starting  point,  which  has  a known  elevation 
of  20  ft.  The  instrument  is  set  at  B,  leveled  up  and  sighted 
to  a rod  held  at  A.  The  target  being  set,  the  reading,  8.42  ft., 
called  a backsight , is  the  distance  that  the  point  where  the 
line  of  sight  cuts  the  rod  is  above  the  point  A , and 
is  to  be  added  to  the  elevation  of  the  point  A.  20.00  -f  8.42 
— 28.42  is  called  the  height  of  instrument  and  is  designated 
by  H.  I.  The  instrument  being  turned  in  the  opposite  direc- 
tion, a point  C is  chosen,  which  must  be  below  the  line  of 
sight.  This  point  is  called  a turning  point , and  is  designated 
by  the  abbreviation  T.  P.  Drive  a peg  at  (7,  or  take  for  a turn- 
ing point  a point  of  rock  or  some  other  permanent  object 
upon  which  the  rod  is  held.  The  reading  at  this  point  is  a 
foresight,  and  is  to  be  subtracted  from  the  height  of  the 
instrument  at  B to  find  the  elevation  of  the  point  at  C. 

Let  the  rod  reading  be  1.20  ft.  As  this  reading  is  a fore- 
sight, it  must  be  subtracted  from  28.42,  the  height  of  instru- 
ment at  B ; 28.42  — 1.20  = 27.22  ft.,  the  elevation  of  the  point 
C.  The  leveler  carries  the  instrument  to  B,  which  should  be 
of  such  a height  above  C that,  when  leveled  up,  the  line  of 
sight  will  cut  the  rod  near  the  top.  The  backsight  to  C gives 
a reading  of  11.56  ft.,  which,  added  to  27.22  ft.,  the  elevation 


LEVELING. 


293 


of  C , gives  38.78  ft.,  the  height  of 
the  instrument  at  D,  The  rodman 
then  goes  to  E,  a point  where  a 
foresight  reading  is  1.35,  which, 
subtracted  from  38.78,  the  H.  I.  at 

D,  gives  37.43  ft.,  the  elevation  of 

E.  The  level  is  then  set  up  at  F \ 
being  careful  that  line  of  sight 
shall  clear  the  hill  at  L.  The  back- 
sight, 6.15  ft.,  added  to  37.43  ft.,  the 
elevation  of  E,  gives  43.58  ft.,  the 
H.  I.  at  F.  The  rod  held  at  G gives 
a foresight  of  10.90  ft.,  which,  sub- 
tracted from  43.58  ft.,  the  H.  I.  at 

F,  gives  32.68  ft.,  the  elevation  at 

G.  Again  moving  the  level  to  H, 
the  backsight  to  G of  4.39  ft.  added 
to  32.68  ft.,  the  elevation  of  G, 
gives  37.07  ft.,  the  H.  I.  at  H.  Hold- 
ing the  rod  at  K , a foresight  of  5.94, 
subtracted  from  37.07,  gives  31.13, 
the  elevation  of  the  point  K.  The 
elevation  of  the  starting  point  A is 
20.00  ft.;  the  elevation  of  the  point 
K is  found  by  direct  leveling  to  be 

31.13  ft.,  and  the 
difference  i n the 
elevations  of  A and 
K is  31.13  — 20.00  = 

11.13  ft.;  that  is,  the 
point  K is  11.13  ft. 
higher  than  the 
point  A. 

Turning  points 
previously  men- 
tioned are  the 
points  where  back- 
si  ghts  and  fore- 
sights are  taken.  The  backsights  are  plus  ( + ) readings,  and 


294 


SURVEYING. 


are  to  be  added;  the  foresights  are  minus  (— ) readings,  and 
are  to  be  subtracted.  A point  for  a foresight  having  been 
determined,  the  rodman  drives  a peg  firmly  in  the  ground 
and  holds  the  rod  upon  it.  After  the  instrument  is  moved, 
set  up,  and  a backsight  taken,  the  peg  is  pulled  up  and  carried 
in  the  pocket  until  another  turning  point  is  called  for.  Turn- 
ing points  should  be  taken  at  about  equal  distances  from  the 
instrument,  in  order  to  equalize  any  small  errors  in  adjust- 
ment. In  smooth  country  an  ordinary  level  will  permit  of 
sights  of  from  300  to  ,500  ft. 

To  Keep  Level  Notes.— Many  forms  are  used.  The  distin- 
guishing feature  of  one  of  the  best  (see  page  295)  is  a single 
column  for  all  rod  readings.  The  backsights  being  additive 
and  the  foresights  subtractive  readings,  they  are  distinguished 
from  other  rod  readings  by  the  characteristic  signs  + (plus) 
and  — (minus).  The  turning  points,  whose  foresight  read- 
ings are  — , are  further  abbreviated  T.  P. 

To  Check  Level  Notes.— A well-known  method  of  checking 
level  notes  provides  for  checking  the  elevations  of  turning 
points  and  heights  of  instrument  only,  which  is  sufficient,  as 
all  other  elevations  are  deduced  from  them.  The  method 
depends  on  the  fact  that  all  backsights  are  additive  (i.  e.  +) 
quantities,  and  all  foresights  are  subtractive  (i.e.—)  quantities. 
The  notes  given  on  page  295  are  checked  as  follows:  The  ele- 
vation of  the  bench  mark  at  station  0 is  100.00  ft.,  to  which  all 
backsights,  or  + readings,  are  to  be  added  and  from  this  sum 
all  foresights,  or  — readings,  are  to  be  subtracted.  The  sum 
of  the  backsights,  with  elevation  of  bench  mark  at  0,  is  122.59. 
Sum  of  foresights  is  24.27,  and  difference  is  98.32  ft.,  the  eleva- 


Thus, 


+ 

100.00 
5.61 
5.41 
11.57 
122.59 
' 24.27 
”~9&32 


10.22 

2.52 

11.53 

24.27 


tion  of  the  turning  point  last  taken.  As 
soon  as  a page  of  level  notes  is  filled, 
the  notes  should  be  checked  and  a 
check  mark  y'  placed  at  the  last  height 
of  instrument  or  elevation  checked. 
When  the  work  of  staking  out  or  cross- 
sectioning  is  being  done,  the  levels 
should  be  checked  at  each  bench 
mark  on  the  line.  After  each  day’s 


work,  the  leveler  must  check  on  the  nearest  bench  mark. 


LEVELING. 


295 


2. 

Date. 

o 

c3 

CO 

t-4 

© 

p< 

a 

p 

CO 

Remarks. 

On  root  of  white  oak 

Spring  Brook. 

Fill. 

Cut. 

Grade. 

Eleva- 

tion. 

o 

o 

o 

o 

99.5  1 

CO 

CO 

05 

<N 

05 

© 

05 

05 

CO 

>d 

05 

1 94.5  1 

96.6 

CO 

05 

CO 

s 

$ 

q 

CO 

o 

CO 

o 

I> 

05 

05 

98.32 

Ht. 

Instru- 

ment. 

105.61 

SB 

© 

o 

rH 1 

iO 

00 

05 

o 

Rod 

Read- 

ing. 

+ 5.61 

«d 

CO 

00 

9.2  1 

1 —10.22 

id 

+ 

CO 

© 

<N 

iO 

iO 

ci 

1 

iO 

+ 

<M 
< 6 

iO 

CO 

o 

—11.53 

1. 

Station 

o 

- 

<N 

CO 

p4 

Eh' 

iO 

5 + 50  I 

T.  P. 

© 

O0 

Ph 

Eh 

296 


SURVEYING. 


Profiles.— A profile  represents  a longitudinal  projection  of 
the  line  of  survey.  In  it  all  abrupt  changes  in  elevation  are 
clearly  outlined.  Vertical  and  horizontal  measurements  are 
usually  represented  by  different  scales,  to  render  irregulari- 
ties of  surface  more  distinct  through  exaggeration.  For 
railroad  work,  profiles  are  commonly  made  to  the  following 
scales,  viz.,  horizontal,  400  ft.  = 1 in.;  vertical,  20  ft.  = 1 in. 

A section  of  profile  paper  is  shown  in  the  following 
diagram.  Every  fifth  horizontal  line  and  every  tenth 
vertical  line  is  heavy.  By  the  aid  of  these  heavy  lines, 
distances  and  elevations  are  quickly  and  correctly  estimated 
and  the  work  of  platting  greatly  facilitated.  The  level  notes 


given  in  the  preceding  diagram  are  platted  in  the  accom- 
panying section.  The  elevation  of  some  horizontal  line  is 
assumed.  This  elevation  is,  of  course,  referred  to  the  datum 
plane,  and  is  the  base  from  which  the  other  elevations  are 
estimated.  Every  tenth  station  number  is  written  at  the 
bottom  of  the  sheet  under  the  heavy  vertical  lines.  The 
profile  is  first  platted  in  pencil  and  then  inked  in  in  black. 

Grade  Lines.— The  principal  use  of  a profile  is  to  enable  the 
engineer  to  establish  a grade  line , i.  e.,  a line  showing  the 
slope  of  the  road  on  which  the  amounts  of  excavation  and 
embankment  depend.  The  rate  of  a grade  line  is  measured 
by  the  vertical  rise  or  fall  in  each  hundred  feet  of  its  length, 


RADII  AND  DEFLECTIONS. 


297 


and  is  designated  by  the  term  per  cent.  Thus,  a grade  line 
that  rises  or  falls  1 ft.  in  each  hundred  feet  of  its  length  is 
called  an  ascending  or  descending  1 per  cent,  grade,  and  is 
written  + 1.0  or  — 1.0  per  hundred.  A rise  or  fall  of  I ft.  in 
each  hundred  feet  is  called  a 0.5  grade,  and  is  written  + 0.5 
or  — 0.5  per  hundred.  The  grade  line  having  been  decided 
on,  it  is  drawn  in  red  ink. 

Example.— The  elevation  of  station  20  is  140.0  ft.;  between 
stations  20  and  100  there  is  an  ascending  grade  of  75 }.  What 
is  the  elevation  of  the  grade  at  station  71? 

Solution. — To  obtain  the  elevation  of  the  grade  at  sta- 
tion 71,  we  add  to  the  elevation  of  the  grade  at  station  20,  or 
140  ft.,  the  total  rise  in  grade  between  stations  20  and  71. 
Accordingly,  71  — 20  = 51;  .75  ft.  X 51  = 38.25  ft.;  140  ft.  + 
38.25  ft.  = 178.25  ft.,  the  elevation  of  grade  at  station  71. 


RADII  AND  CHORD  AND  TANGENT 
DEFLECTIONS. 


The  formulas  used  in  the  computation  of  the  following 
table  are  as  follows: 


For  radius,  R = - — -. 

sin  D 

C2 

For  chord  deflection,  d = — . 

ti 

c2 

For  tangent  deflection,  tan  deflection  = — — 

2 R 


In  these  formulas,  R is  the  radius  of  the  curve,  D is  its 
deflection  angle  (equal  to  one-half  the  degree  of  curve),  and 
c is  the  length  of  chord  for  which  the  chord  or  tangent 
deflection  is  to  be  determined.  The  chord  and  tangent  deflec- 
tions given  in  the  table  are  computed  for  chords  of  100  feet. 

Thus,  for  a 6°  curve  the  deflection  angle  is  3°,  the  sine  of 
which  is  .052336.  Hence,  for  the  radius  and  chord  deflection, 
we  have 


R = 


50 


= 955.37  ft. 


d = 


.052336 

as  given  in  the  table.  The  tangent  deflection  is  always 
one-half  the  chord  deflection. 


298 


SURVEYING. 


Table  of  Radii  and  Deflections. 


d 

g 

d 

d 

6 

■gS 

g.2 

6 

■a  .2 

d.2 

a>+3 
fcoo 
p a> 

h 

c3 

o U 
_d  © 

Soy 

a © 

S> 

*d 

o o 
Jh  © 

a> 

ft 

ft 

^ 0) 
ft 

S«S 

r_|  © 

Hft 

a> 

ft 

ft 

w © 
ft 

£_j  <D 

Hft 

o / 
0 5 

68,754.94 

.145 

.073 

o / 

3 25 

1,677.20 

5.962 

2.981 

10 

34,377.48 

.291 

.145 

30 

1,637.28 

6.108 

3.054 

15 

22,918.33 

.436 

.218 

35 

1,599.21 

6.253 

3.127 

20 

17,188.76 

.582 

.291 

40 

1,562.88 

6.398 

3.199 

25 

13,751.02 

.727 

.364 

45 

1,528.16 

6.544 

3.272 

30 

11,459.19 

.873 

.436 

50 

1,494.95 

6.689 

3.345 

35 

9,822.18 

1.018 

.509 

55 

1,463.16 

6.835 

3.417 

40 

8,594.41 

1.164 

.582 

45 

7,639.49 

1.309 

.654 

4 0 

1,432.69 

6.980 

3.490 

50 

6,875.55 

1.454 

.727 

5 

1,403.46 

7.125 

3.563 

55 

6,250.51 

1.600 

.800 

10 

1,375.40 

7.271 

3.635 

15 

1,348.45 

7.416 

3.708 

1 0 

5,729.65 

1.745 

.873 

20 

1,322.53 

7.561 

3.781 

5 

5,288.92 

1.891 

.945 

25 

1,297.58 

7.707 

3.853 

10 

4,911.15 

2.036 

1.018 

30 

1,273.57 

7.852 

3.926 

15 

4,583.75 

2.182 

1.091 

35 

1,250.42 

7.997 

3.999 

20 

4,297.28 

2.327 

1.164 

40 

1,228.11 

8.143 

4.071 

25 

4,044.51 

2.472 

1.236 

45 

1,206.57 

8.288 

4.144 

30 

3,819.83 

2.618 

1.309 

50 

1,185.78 

8.433 

4.217 

35 

3,618.80 

2.763 

1.382 

55 

1,165.70 

8.579 

4.289 

40 

3,437.87 

2.909 

1.454 

45 

3,274.17 

3.054 

1.527 

5 0 

1,146.28 

8.724 

4.362 

50 

3,125.36 

3.200 

1.600 

5 

1,127.50 

8.869 

4.435 

55 

2,989.48 

3.345 

1.673 

10 

1,109.33 

9.014 

4.507 

15 

1,091.73 

9.160 

4.580 

2 0 

2,864.93 

3.490 

1.745 

20 

1,074.68 

9.305 

4.653 

5 

2,750.35 

3.636 

1.818 

25 

1,058.16 

9.450 

4.725 

10 

2,644.58 

3.781 

1.891 

30 

1,042.14 

9.596 

4.798 

15 

2,546.64 

3.927 

1.963 

35 

1,026.60 

9.741 

4.870 

20 

2,455.70 

4.072 

2.036 

40 

1,011.51 

9.886 

4.943 

25 

2,371.04 

4.218 

2.109 

45 

996.87 

10.031 

5.016 

30 

2,292.01 

4.363 

2.181 

50 

982.64 

10.177 

5.088 

35 

2,218.09 

4.508 

2.254 

55 

968.81 

10.322 

5.161 

40 

45 

2,148.79 

2,083.68 

4.654 

4.799 

2.327 

2.400 

6 0 

955.37 

10.467 

5.234 

50 

55 

2,022.41 

1,964.64 

4.945 

5.090 

2.472 

2.545 

5 

10 

15 

942.29 

929.57 

917.19 

10.612 

10.758 

10.903 

5.306 

5.379 

5.451 

3 0 

1,910.08 

5.235 

2.618 

20 

905.13 

11.048 

5.524 

5 

1,858.47 

5.381 

2.690 

25 

893.39 

11.193 

5.597 

10 

1,809.57 

5.526 

2.763 

30 

881.95 

11.339 

5.669 

15 

1,763.18 

5.672 

2.836 

35 

870.79 

11.484 

5.742 

20  J 

1.719.12 

5.817 

2.908 

40 

859.92 

11.629 

5.814 

RADII  AND  DEFLECTIONS. 


299 


Table— ( Continued). 


Degree. 

Radii. 

Chord 

Deflection. 

Tangent 

Deflection. 

Degree. 

Radii. 

Chord 

Deflection. 

Tangent 

Deflection. 

o / 

6 45 

849.32 

11.774 

5.887 

o 

10 

/ 

0 

573.69 

17.431 

8.716 

50 

838.97 

11.919 

5.960 

10 

564.31 

17.721 

8.860 

55 

828.88 

12.065 

6.032 

20 

555.23 

18.011 

9.005 

30 

546.44 

18.300 

9.150 

7 0 

819.02 

12.210 

6.105 

40 

537.92 

18.590 

9.295 

5 

809.40 

12.355 

6.177 

50 

529.67 

18.880 

9.440 

10 

800.00 

12.500 

6.250 

15 

790.81 

12.645 

6.323 

11 

0 

521.67 

19.169 

9.585 

20 

781.84 

12.790 

6.395 

10 

513.91 

19.459 

9.729 

25 

773.07 

12.936 

6.468 

20 

506.38 

19.748 

9.874 

30 

764.49 

13.081 

6.540 

30 

499.06 

20.038 

10.019 

35 

756.10 

13.226 

6.613 

40 

491.96 

20.327 

10.164 

40 

747.89 

13.371 

6.685 

50 

485.05 

20.616 

10.308 

45 

739.86 

13.516 

6.758 

50 

732.01 

13.661 

6.831 

12 

0 

478.34 

20.906 

10.453 

55 

724.31 

13.806 

6.903 

10 

471.81 

12.195 

10.597 

20 

465.46 

21.484 

10.742 

8 0 

716.78 

13.951 

6.976 

30 

459.28 

21.773 

10.887 

5 

709.40 

14.096 

7.048 

40 

453.26 

22.063 

11.031 

10 

702.18 

14.241 

7.121 

50 

447.40 

22.352 

11.176 

15 

695.09 

14.387 

7.193 

20 

688.16 

14.532 

7.266 

13 

0 

441.68 

22.641 

11.320 

25 

30 

681.35 

674.69 

14.677 

14.822 

7.338 

7.411 

10 

20 

436.12 

430.69 

22.930 

23.219 

11.465 

11.609 

35 

668.15 

14.967 

7.483 

30 

425.40 

23.507 

11.754 

40 

661.74 

15.112 

7.556 

40 

420.23 

23.796 

11.898 

45 

655.45 

15.257 

7.628 

50 

415.19 

24.085 

12.043 

50 

649.27 

15.402 

7.701 

55 

643.22 

15.547 

7.773 

14 

0 

410.28 

24.374 

12.187 

9 0 

637.27 

15.692 

7.846 

10 

405.47 

24.663 

12.331 

5 

631.44 

15.837 

7.918 

20 

400.78 

24.951 

12.476 

10 

625.71 

15.982 

7.991 

30 

396.20 

25.240 

12.620 

15 

620.09 

16.127 

8.063 

40 

391.72 

25.528 

12.764 

20 

614.56 

16.272 

8.136 

50 

387.34 

25.817 

12.908 

25 

609.14 

16.417 

8.208 

30 

603.80 

16.562 

8.281 

15 

0 

383.06 

26.105 

13.053 

35 

598.57 

16.707 

8.353 

10 

378.88 

26.394 

13.197 

40 

593.42 

16.852 

8.426 

20 

374.79 

26.682 

13.341 

45 

588.36 

16.996 

8.498 

30 

370.78 

26.970 

13.485 

50 

583.38 

17.141 

8.571 

40 

366.86 

27.258 

13.629 

55 

578.49 

17.286 

8.643 

50 

363.02 

27.547 

13.773 

300 


SURVEYING. 


Table — ( Continued). 


Degree. 

Radii. 

Chord 

Deflection. 

Tangent 

Deflection. 

Degree. 

Radii. 

Chord 

Deflection. 

Tangent 

Deflection. 

o / 

o / 

16  0 

359.26 

27.835 

13.917 

18  10 

316.71 

31.574 

15.787 

10 

355.59 

28.123 

14.061 

20 

313.86 

31.861 

15.931 

20 

351.98 

28.411 

14.205 

30 

311.06 

32.149 

16.074 

30 

348.45 

28.699 

14.349 

40 

308.30 

32.436 

16.218 

40 

344.99 

28.986 

14.493 

50 

305.60 

32.723 

16.361 

50 

341.60 

29.274 

14.637 

19  0 

302.94 

33.010 

16.505 

17  0 

338.27 

29.562 

14.781 

10 

300.33 

33.296 

16.648 

10 

335.01 

29.850 

14.925 

20 

297.77 

33.583 

16.792 

20 

331.82 

30.137 

15.069 

30 

295.25 

33.870 

16.935 

30 

328.68 

30.425 

15.212 

40 

292.77 

34.157 

17.078 

40 

325.60 

30.712 

15.356 

50 

290.33 

34.443 

17.222 

50 

322.59 

31.000 

15.500 

18  0 

319.62 

31.287 

15.643 

20  0 

287.94 

34.730 

17.365 

RETAINING  WALLS. 

On  the  Theory  of  Retaining  Walls.— Let  abdc , Fig.  1,  be  a 
retaining  wall  with  battered  face  and  vertical  back.  The  top 
b e of  the  backing  is  level  with  the  top  of  the  wall.  Let  d e 
represent  the  natural  slope  of  the  material  composing  the 
filling,  viz.,  1£  horizontal  to  1 vertical,  which  is  the  average 
of  materials  used  for  back  filling. 

It  is  assumed  that  the  wall  abdc  is  heavy  enough  to  resist 
sliding  along  its  base  and  that  it  can  fail  only  by  overturning, 
i.  e.,  rotating  about  its  toe  c. 
Now,  if  the  angle  ode  (between 
the  vertical  line  o d drawn  from 
the  inner  bottom  edge  of  the 
wall  and  the  natural  slope  de) 
be  bisected  by  the  line  df,  the 
angle  o df  is  called  the  angle , 
and  the  line'd  f the  slope,  of  maxi- 
mum pressure.  The  triangular  prism  of  earth  o df  is  called 
the  prism  of  maximum  pressure , because,  if  considered  as  a 


Fig.  1. 


RETAINING  WALLS. 


301 


wedge  acting  against  the  back  of  the  wall,  it  would  exert  a 
greater  pressure  against  it  than  would  the  entire  triangle  ode 
of  earth  considered  as  a single  wedge.  For  though  the  latter 
is  more  than  double  the  weight  of  the  former,  yet  it  receives 
much  greater  support  from  the  underlying  earth.  It  has  been, 
proved  by  experiment  that,  if  the  triangle  of  earth  ode  is 
divided  by  any  line  df  into  wedges,  the  wedge  that  will 
press  most  against  the  wall  is  that  formed  wThen  the  line  df 
divides  the  angle  ode  into  two  equal  parts. 

The  angle  odh  formed  by  the  vertical  o d and  the  hori- 
zontal dh  is  90°.  The  angle  of  natural  slope  hde  is  33° 41'; 
hence,  the  angle  odf  of  maximum  pressure  is  equal  to 
(90°  — 33°  41')  h-  2 = 28°  09'. 

In  making  calculations,  only  one  foot  of  the  length  of  wall 
and  of  the  backing  is  taken,  so  all  that  is  necessary  is  to  take 
the  area  of  the  section  of  the  wall  and  backing.  The  mate- 
rial composing  the  backing  is  supposed  to  be  perfectly  dry 
and  to  possess  no  cohesive  power,  which  is  practically  true  of 
pure  sand. 

If  we  conceive  the  wall  abdc , Fig.  J,  to  be  suddenly 
removed,  the  triangle  bdf  of  sand  included  between  the  line 
of  maximum  pressure  df  and  the  vertical  back  b d of  the  wall 
would  slide  downward,  impelled  by  a force  n P,  acting  in  a 
direction  nP  at  right  angles  to  the  side  b d of  the  triangle, 
i.  e.,  at  right  angles  to  the  vertical  back  bd  of  the  wall;  the 
center  of  pressure  being  at  P one-third  of  the  distance  between 
b and  d measured  from  the  bottom  of  the  wall  d.  The 
amount  of  this  force  n P is: 

Perpendicular  pressure  = Wt  of  triangle,°f  ear*h  b,df*  0/. 

vertical  depth  o d 

This  formula  not  only  applies 
to  walls  with  vertical  backs, 
as  in  Fig.  1,  but  to  those  with 
inclined  backs,  as  in  Fig.  2,  for 
inclinations  as  high  as  6 in.  hori- 
zontal to  1 ft.  vertical,  which 
is  rarely  met  with  and  never 
exceeded. 


Fig.  2. 


Friction  Caused  by  Pressure  of  Backing.— If  all  the  backing 


302 


SURVEYING. 


material  contained  between  the  line  of  natural  slope  and  the 
back  of  the  wall  were  unconfined,  it  would  slide,  producing 

motion;  but 
confined  by 
the  retaining 
wall,  the 
force  is  con- 
verted into 
pressure  of 
earth  against 
the  back  of 
the  wall,  re- 
sisted by  the  friction  between  the  compressed 
earth  and  the  wall. 

If  the  wall  were  to  begin  to  overturn 
about  its  toe  c (Figs.  1 and  2)  as  a fulcrum, 
its  back  bd  "would  rise,  producing  friction 
against  the  backing.  So  long  as  the  wall 
does  not  move,  the  friction  of  the  backing 
acts  constantly,  and  must,  therefore,  be  one 
of  the  forces  that  prevent  overturning.  We 
ascertain  the  amount  and  effect  of  this  fric- 
Let  abdc,  Fig.  3,  be  a retaining  wall,  and  let 
n P represent  to  some  scale  the  perpendicular  pressure  against 
the  back  of  the  wall  calculated  by  the  preceding  formula, 
viz0,  perpendicular  pressure  — 

weight  of  triangle  dbfx  of 
vertical  depth  o d 
Make  the  angle  n Ph  equal  to  the  angle  of  wall  friction,  viz., 
that  at  which  a plane  of  masonry  must  be  inclined  to  the  hori- 
zontal in  order  that  dry  sand  and  earth  may  slide  freely 
over  it,  and  taken  at  33°  41'.  Draw  nh  perpendicular  to  nP 
and  complete  the  parallelogram  nhkP.  Then  will  k P repre- 
sent to  the  same  scale  the  amount  of  friction  against  the  back  of 
the  wall.  As  the  friction  acts  in  the  direction  of  the  back 
bd  of  the  wall,  it  may  be  considered  as  acting  at  any  point 
Pof  the  line  of  the  back,  and  we  will  have  two  forces,  viz., 
the  perpendicular  pressure  nP  and  the  friction  kP  acting 
at  P.  By  composition  and  resolution  of  forces,  the  diagonal 


Fig.  3. 

tion  as  follows: 


nP  = ■ 


RETAINING  WALLS. 


303 


h P measured  to  the  same  scale  will  give  us  the  amount  of 
their  resultant,  which  is  approximately  the  single  theoretical 
force  both  in  amount  and  direction  that  the  wall  has  to  resist. 
This  force  includes  the  wall  friction.  The  force  h P is  always 
equal  to  the  perpendicular  force  n P , divided  by  the  cosine  of 
the  angle  of  wall  friction.  The  cosine  of  the  angle  of  wall 
friction  is  .832  and  the  value  of  the  force  h P may  be  expressed 
in  the  following  formula: 

Approximate  theoretical  pressure 

_ hp__  weight  of  triangle  bdfX  of 
~ ~ vertical  height  odx  .832 

When  the  back  of  the  wall  does  not  incline  forward  more 
than  6 in.  horizontal  to  1 ft.  vertical,  equal  to  an  angle  of 
about  26°  34',  the  following  formula  by  Trautwine  is  used,  viz.: 
Approximate  theoretical  pressure 

= hP  = weight  of  triangle  bdfX  .643, 
which  includes  friction  of  earth  against  the  back  of  the  wall. 

To  Find  the  Overturning  and  Resisting  Forces. — To  find  the 
overturning  tendency  of  the  earth  pressure  and  the  resistance  of 
the  wall  against  being  overturned  about  its  toe  c,  as  a fulcrum 
(see  Fig.  3).  Find  the  center  of  gravity  g of  the  wall,  and 
through  g draw  the  vertical  line  gi.  Produce  the  line  of 
pressure  h P,  and  draw  c v at  right  angles  to  this  line.  To  any 
convenient  scale,  lay  off  1 1 equal  to  the  weight  of  the  wall 
and  to  the  same  scale  Im  equal  to  the  pressure  hP.  Com- 
plete the  parallelogram  Imst.  The  diagonal  l s will  be  the 
resultant  of  the  pressure  and  the  weight  of  the  wall.  The 
stability  of  the  wall  will  increase  as  the  distance  cr  from 
the  toe  to  the  point  where  the  resultant  Is  cuts  the  base, 
increases.  To  insure  stability,  c r must  be  greater  than  \cd. 

The  pressure  h P,  if  multiplied  by  its  leverage  c v,  will  give 
the  moment  of  the  pressure  about  c,  and  the  weight  of  the 
wall  l ty  multiplied  by  its  leverage  c r',  will  give  the  moment 
of  the  wall.  The  wall  is  secure  against  overturning  in  pro- 
portion as  its  moment  exceeds  that  of  the  pressure. 

For  example,  let  the  height  of  the  wall  abdc,  in  Fig.  3, 
be  9 ft.;  the  thickness  at  the  base  c d,  4.5ft.,  and  at  the  top  a b , 
2 ft.;  and  the  batter  of  a c be  1 in.  to  the  foot.  The  triangle  of 
earth  b df  has  a base  bf  = 6.57  ft.  and  altitude  do  = 9 ft 


304 


SURVEYING. 


Taking  the  section  as  1 ft.  in  thickness,  we  have  the  con- 
tents equal  to  6.57  X 9 -r-  2 = 29.56  cu.  ft.  Assuming  the 
material  to  weigh  120  lb.  per  cu.  ft.,  the  weight  of  the  triangle 
bdf  is  29.56  X 120  = 3,547  lb.;  of  = 4.81  ft.  3,547  X 4.81 
= 17,061.  17,061  -7-  od  = 1,895.7  lb.  = the  perpendicular 

pressure  n P Lay  off  on  a line  perpendicular  to  the  back  of 
the  wall  at  P,  to  a scale  of  2,000  lb.  = 1 in.,  nP  = 1,895.7  -4- 
2,000  = .948  in.,  the  perpendicular  pressure.  Draw  Ph> 
making  the  angle  nPh  — 33°  41'.  Draw  n h intersecting  h P 
in  h ; then  will  n h to  the  same  scale  equal  the  friction  of  the 
earth  against  the  back  of  the  wall.  Completing  this  parallelo- 
gram, nhkP,  the  diagonal  hP  = 1,139  in.,  which,  to  a 
scale  of  2,000  lb.  = 1 in.,  amounts  to  2,278  lb.,  and  is  the 
resultant  of  the  pressure  and  the  friction. 

Produce  the  resultant  h P to  u.  We  next  find  the  center 
of  gravity  g of  the  wall  abdc.  The  section  of  the  wall  is  a 
trapezoid,  and  the  center  of  gravity  g is  readily  found  as 
follows:  Produce  the  upper  base  of  the  section  to  x , and 
make  ax  = cd  = 4.5  ft.  Then  produce  the  lower  base  in 
the  opposite  direction  to  y,  and  make  d y — a b = 2 ft.  Join 
x and  y.  Find  the  middle  points  x'  and  y'  of  the  upper  and 
lower  bases  of  the  section.  Join  these  points.  The  inter- 
section g of  the  lines  x y and  x'  y'  is  the  center  of  gravity  of 
the  trapezoid  abdc. 

The  volume  of  the  section  of  wall  abdc  is  readily  found. 
The  sum  of  top  and  bottom  widths  = 2.0  + 4.5  = 6.5  ft. 
6.5  -r-  2 = 3.25  ft.  3.25  X 9 = 29.25  cu.  ft.  29.25  X 154  = 4,504  lb. 
(the  weight  per  cubic  foot  of  good  mortar  rubble  = 154  lb.) 
= the  weight  of  the  section  abdc.  Draw  through  g a verti- 
cal line  g i,  and  lay  off  on  it,  to  a scale  of  2,000  lb.  to  the  inch, 
from  the  point  l,  where  the  line  of  gravity  intersects  the  pro- 
longation of  the  line  of  pressure  h P,  the  length  1 1 equal  to 
4,504  lb.,  the  weight  of  the  wall.  Lay  off  from  l on  the  pro- 
longation of  hP,  Im  equal  to  2,278  lb.  to  the  same  scale. 
Complete  the  parallelogram  Imst.  The  diagonal  Is  represents 
the  resultant  of  the  pressure  and  of  the  weight  of  the  wall. 
The  distance  c r from  the  toe  c to  the  intersection  of  the 
resultant  l s with  the  base  c d is  more  than  one-third  of  the 
width  of  the  base,  which  insures  ample  stability. 


TUNNELS. 


305 


Fig.  4. 


Pressure  of  the  Backing  on  Surcharged  Walls.— In  Fig. 4 the 
surcharge  of  backing  mbo  slopes  from  b at  its  natural  slope, 
and  attains  its  maximum 
pressure  where  the  slope 
of  maximum  pressure  d k 
intersects  the  natural 
slope  b m at  /.  Any  addi- 
tional height  of  sur- 
charge does  not  increase 
this  pressure.  If  the  sur- 
charge slopes  from  a,  as 
shown  by  the  line  ap , or 
from  any  point  between 
a and  b , then  the  slope  of 
maximum  pressure  must 
be  extended,  intersecting 
the  slope  from  a in  the 
point  k.  The  prism  of  maximum  pressure  will  then  be  dik. 
The  triangle  of  earth  abi  on  the  top  of  the  wall  exerts 
no  pressure  against  the  back  of  the  wall,  but  adds  to  its 
stability. 

Having  found  the  weight  of  the  triangle  bdf  we  have 
approximate  pressure  — weight  of  triangle  bdfX  .643, 
which  includes  the  pressure  of  the  backing  and  the  friction 
of  the  earth  against  the  back  of  the  wall. 

Draw  Pn  perpendicular  to  the  back  of  the  wall  and  draw 
hP  making  the  angle  nPh  = 33°  41,  the  angle  of  wall 
friction.  Then,  hP  will  be  the  direction  of  the  pressure. 
The  point  of  application  of  this  pressure  will  not  always  be 
at  P,  one-third  of  the  height  of  bd  measured  from  d , but 
above  P,  as  at  r,  where  a line  drawn  from  the  center  of 
gravity  g of  the  prism  of  maximum  pressure  dik  (omitting  any 
earth  resting  directly  upon  the  top  of  the  wall),  and  parallel 
to  the  line  d k of  maximum  pressure,  cuts  the  back  b d of  the 
wall.  The  center  of  pressure  P will  be  at  one-third  the 
height  of  the  wall  when  the  sustained  earth  dbs  or  dbf 
forms  a complete  triangle , one  of  whose  angles  is  at  b,  the  inner 
top  edge  of  the  wall.  For  all  other  surcharges,  the  point  of 
pressure  will  be  above  P. 


306 


SURVEYING. 


TUNNEL  SECTIONS. 

Tunnel  sections  vary  somewhat,  according  to  the  material 
to  be  excavated,  but  the  general  form  and  dimensions  are 
much  the  same. 


Section  of  Section  of 

Double-Track  Tunnel . Single-Track  Tunnel. 


Fig.  1.  Fig.  2. 


The  general  dimensions  are  as  follows:  For  double  track, 
from  22  to  27  ft.  wide  and  from  21  to  24  ft.  high,  and  for  single 
track,  from  14  to  16  ft.  wide  and  from  17  to  20  ft.  high  (see 
Figs.  1 and  2). 

In  seamy  or  rotten  rock  the  section  is  sufficiently  enlarged 
to  receive  a lining  of  substantial  rubble  or  brick  masonry 
laid  in  good  cement  mortar.  When  the  material  has  not 
sufficient  consistency  to  sustain  itself  until  the  masonry  lining 
is  built,  resort  is  had  to  timbering,  which  furnishes  the 
necessary  support.  

CALCULATION  OF  EARTHWORK. 

In  calculating  the  quantity  of  material  in  excavation  and 
embankment,  two  general  methods  are  used,  namely,  the 
end-area  formula  and  the  prismoidal  formula. 

Calculation  by  the  end-area  method  consists  in  multiply- 
ing the  mean,  or  average,  area  in  square  feet  of  two  consecu- 
tive sections  by  the  distance  in  feet  between  them.  Thus, 


EARTHWORK. 


307 


let  A represent  the  area  in  square  feet  of  one  section;  B, 
the  area  in  square  feet  of  the  next  section;  C,  the  number 
of  feet  between  the  sections;  and  D,  the  total  number  of  cubic 
feet  in  the  prismoid  lying  between  these  sections.  Then, 

A + B 

D = — - — X C , approximately. 

The  distance  between  sections  should  not  be  more  than 
100  ft.,  and  should  be  less  if  the  surface  of  the  ground  is 
irregular. 

A more  accurate  result  is  obtained  by  the  use  of  the  pris- 
moidal  formula.  In  applying  the  prismoidal  formula  to  the 
calculation  of  cubic  contents,  it  is  requisite  to  know  the 
middle  cross-section  between  each  two  that  are  measured  on 
the  ground.  The  dimensions  of  this  middle  section  are  the 
means  of  the  dimensions  of  the  end  sections. 

Calling  one  of  the  given  sections  A,  the  other  B,  the  mid- 
dle (not  the  mean)  section  M,  the  distance  between  the 
sections  L,  and  the  required  contents  St  we  have,  by  the 
prismoidal  formula, 

S=§M  + 4M+B). 

Example.— Two 
sections  are  repre- 
sented by  Figs.  1 
and  2,  and  are  de- 
noted by  the  letters 
A and  B.  The  per- 
pendicular  dis- 
tance between 
them  is  50  ft.  It 
is  required  to  find  the  cubical  contents  of  the  prismoid. 

Solution.— The  sec- 
tion given  in  Fig.  1 is 
composed  of  the  four 
triangles  a,  b,  c , and  d. 
The  triangles  a and  b 
have  equal  bases  of  9 ft. , 
the  half  width  of  the 
Fig.  2.  roadway;  hence,  if  we 


J— i ZL& 

£5  * 

\ « 

^ 1 

' — i. 

" j 

t“  ' ^ 

Fig.  1. 


308 


SURVEYING. 


take  half  the  sum  of  their  altitudes  and  multiply  it  by  the 
common  base  we  shall  have  the  sum  of  the  areas  of  the 
triangles  a and  b. 

The  triangles  c and  d have  a common  base  8 ft.,  the  center 
cut  of  the  section,  and  if  we  take  the  half  sum  of  the  side 
distances  and  multiply  it  by  8 ft.,  we  shall  obtain  the  areas  of 
the  triangles  c and  d.  Taking  the  dimensions  of  section  A 
given  in  Fig.  1,  we  have 

12  g _i_  5 

Areas  of  triangles  a + b = — — X 9 = 80.1  sq.  ft. 

Areas  of  triangles  c + d = X 8 = 143.2  sq.  ft. 

Total  area  of  section  A — 223.3  sq.  ft. 

Taking  the  dimensions  of  the  section  B given  in  Fig.  2,  we 
have 

9 7 + 22 

Areas  of  triangles  a'  + b’  = — 1 — - — — X 9 = 53.55  sq.  ft. 

18  7 4-  ii  2 

Areas  of  triangles  c'  + d'  = — : — — — - X 5 = 74.75  sq.  ft. 

Total  area  of  section  B = 128.3  sq.  ft. 

In  applying  the  prismoidal  formula  we  calculate  the  area 
of  a section  midway  between  the  given  sections,  and  for  its 


dimensions  we  take  the  mean  of  the  dimensions  of  the  given 
sections.  These  dimensions  will  be  as  follows  : 

Center  cut,  - ^ 5 = 6.5  ft. 

Right-side  distance,  — = 12.6  ft. 

Left-side  distance,  = 20.25  ft. 


2 


TRACKWORK. 


309 


With  dimensions  thus  found,  construct  the  section  M 
shown  in  Fig.  3. 

The  area  of  section  M is  computed  by  the  same  method 
as  that  used  with  sections  A and  B in  Figs.  1 and  2,  and  is  as 
follows: 

Area  of  triangles  a"+  6"  = 11-2  + X 9 = 66.6  sq.  ft. 

on  o _l  i o fi 

Area  of  triangles  c"+  d,"  — — — - — - X 6.5  = 106.6  sq.  ft. 

Total  area  of  section  M = 173.2  sq.  ft. 
Denoting  the  distance  between  the  sections  by  L and  the 
cubical  contents  of  the  prismoid  by  S , we  have,  by  substi- 
tuting in  the  prismoidal  formula, 

S = |u+4Jf  + £). 

50 

S = -g  (223.3  + 4 X 173.2  + 128.3)  = 8.703  cu.  ft.  = 322.3  cu.  yd. 


TRACKWORK. 

Curving  Rails.— When  laying  track  on  curves,  in  order  to 
have  a smooth  line,  the  rails  themselves  must  conform  to  the 
curve  of  the  center  line.  To  accomplish  this,  the  rails  must 
be  curved.  The  curving  should  be  done  with  a rail  bender 
or  with  a lever,  preferably  with  the  former. 

To  guide  those  in  charge  of  this  work,  a table  of  middle 
and  quarter  ordinates  for  a 30-ft.  rail  for  all  degrees  of  curve 
should  be  prepared. 

The  following  table  of  middle  ordinates  for  curving  rails 
is  calculated  by  using  the  formula 


in  which  m = middle  ordinate; 

c = chord,  assumed  to  be  of  the  same  length  as 
the  rail; 

R = radius  of  the  curve. 

The  results  obtained  by  this  formula  are  not  theoretically 
correct,  yet  the  error  is  so  small  that  it  may  be  ignored  in 
practical  work. 


310 


SURVEYING. 


In  curving  rails,  the  ordinate  is  measured  by  stretching 
a cord  from  end  to  end  of  the  rail  against  the  gauge  side,  as 
shown  in  Fig.  1.  Suppose  the  rail  A B is  30  ft.  in  length,  and 

the  curve  8°.  Then, 
^ ”T  — ^ by  the  previous  prob- 

7 lem>  the  middle 

K * * ordinate  at  a should 

FlG>  be  1§  in.  To  insure 

a uniform  curve  to  the  rails,  the  ordinates  at  the  quarters  b 
and  b'  should  be  tested.  In  all  cases  the  quarter  ordi- 
nates should  be  three-quarters  of  the  middle  ordinate. 
In  Fig.  1,  if  the  rail  has  been  properly  curved,  the  quarter 
ordinates  at  b and  b'  will  be  £ X If  in.  = Iff,  say  1§  in. 


Middle  Ordinates  for  Curving  Rails. 


TRACKWORK. 


311 


In  trackwork  it  is  often  necessary  to  ascertain  the  degree 
of  a curve,  though  no  transit  is  available  for  measuring  it. 
The  following  table  contains  the  middle  ordinates  of  a 1° 
curve  for  chords  of  various  lengths: 

The  lengths  of 
the  chords  are 
varied,  so  that  a 
longer  or  shorter 
chord  may  be  used, 
according  as  the 
curve  is  regular 
or  not. 

The  table  is  ap- 
plied  as  follows: 

Suppose  the  middle 
ordinate  of  a 44-ft. 
chord  is  3 in.  We  find  in  the  table  that  the  middle  ordi- 
nate of  a 44-ft.  chord  of  a 1°  curve  is  £ in.  Hence,  the  degree 
of  the  given  curve  is  equal  to  the  quotient  of  3 £ = 6°  curve. 

Elevation  of  Curves.— To  counteract  the  centrifugal  force 
developed  when  a car  passes  around  a curve,  the  outer  rail  is 
elevated.  The  amount  of  elevation  will  depend  on  the  radius 
of  the  curve  and  the  speed  at  which  trains  are  to  be  run. 
There  is,  however,  a limit  in  track  elevation  as  there  is  a limit 
in  widening  gauge,  beyond  which  it  is  not  safe  to  pass. 

The  best  authorities  on  this  subject  place  the  maximum 
elevation  at  one-seventh  the  gauge,  or  about  8 in.  for  standard 
gauge  of  4 ft.  8£  in.  The  gauge  on  a 10°  curve  elevated  for  a 
speed  of  40  miles  an  hour  should  be  widened  to  4 ft.  9i  in. 

All  curves,  when  possible,  should  have  an  elevated 
approach  on  the  straight  main  track,  of  such  length  that 
trains  may  pass  on  and  off  the  curve  without  any  sudden  or 
disagreeable  lurch. 

A good  rule  for  curve  approaches  is  the  following:  For 
each  half  inch  or  fraction  thereof  of  curve  elevation,  add  30 
ft.,  for  1 rail  length,  to  the  approach;  that  is,  if  a curve  has  an 
elevation  of  2 in.,  the  approach  will  have  as  many  rail  lengths 
as  the  number  of  timqp  \ is  contained  in  2,  or  4.  The  approach 
will,  therefore,  have  a length  of  4 rails  of  30  ft.  each,  or  120  ft. 


Length  of  Chord. 
Feet. 

Middle  Ordinate 
of  a 1°  Curve. 
Inches. 

20 

Va 

30 

M 

44 

y% 

50 

V* 

62 

l 

100 

2/^ 

120 

3% 

312 


SURVEYING. 


The  following  table  for  elevation  of  curves  is  a compromise 
between  the  extremes  recommended  by  different  engineers. 
It  is  a striking  fact  that  experienced  trackmen  never  elevate 
track  above  6 in.  and  many  of  them  place  the  limit  at  5 in. 


Degree 

of 

Curve. 

Length  of 
Approach. 
Feet. 

Elevation. 

Inches. 

Width 

of 

Gauge. 

Speed 
of  Train. 

Miles 
per  Hour. 

1 

60 

1 

4'  834" 

60 

2 

120 

2 

4'  834" 

60 

3 

4 

150 

180 

1 

4'  8%" 
4'  8%" 

60 

55 

5 

180 

3 

4'  8%" 

50 

6 

210 

3)4 

4'  8%" 

45 

7 

210 

3)1 

4'  9" 

40 

8 

240 

3% 

4'  9" 

35 

9 

240 

4 

4'  9" 

30 

10 

270 

4 )4 

4'  9" 

25 

11 

270 

4)| 

4'  9)4" 

20 

12 

270 

4 % 

4'  934" 

15 

13 

240 

4)| 

4'  934" 

10 

14 

240 

434 

4'  9)4" 

10 

15 

240 

4 

4'  9)|" 

10 

16 

240 

4 

4'  934" 

10 

The  Elevation  of  Turnout  Curves.— The  speed  of  all  trains  in 
passing  over  turnout  curves  and  crossovers  is  greatly  reduced, 
so  that  an  elevation  of  ) in.  per  degree  is  amply  sufficient  for 
all  curves  under  16°.  On  curves  exceeding  16°,  the  elevation 
may  be  held  at  4 in.  until  20°  is  reached,  and  on  curves 
extending  20°,  t3«j  in.  of  elevation  per  degree  may  be  allowed 
until  the  total  elevation  amounts  to  5 in.,  which  is  sufficient 
for  the  shortest  curves. 

The  Frog.— The  frog  is  a device  by  means  of  which  the  rail 
at  the  turnout  curve  crosses  the  rail  of  the  main  track.  The 
frog  shown  in  Fig.  2 is  made  of  rails  having  the  same  cross- 
section  as  those  used  in  the  track.  The  wedge-shaped  part  A 
is  the  tongue , of  which  the  extreme  end  a is  the  point.  The 
space  6,  between  the  ends  c and  d of  the  rails,  is  the  mouth , 
and  the  channel  that  they  form  at  its  narrowest  point  e is  the 
throat.  The  curved  ends  / and  g are  the  wings. 


TRACKWORK. 


313 


That  part  of  the  frog  between  A and  A'  is  called  the  heel. 
The  width  h of  the  frog  is  called  its  spread.  Holes  are  drilled 


Fig.  2. 


in  the  ends  of  the  rails  c,  d , k,  and  l to  receive  the  holts  used 
in  fastening  the  rail  splices,  so  that  the  rails  of  which  the 
frog  is  composed  form  a part  of  the  continuous  track. 

The  Frog  Number.— The  number  of  a frog  is  the  ratio  of  its 
length  to  its  breadth;  i.  e.,  the  quotient  of  its  length  divided 
by  its  breadth. 

Thus,  in  Fig.  2,  if  the  length  a'  l,  from  point  to  heel  of  frog 
is  5 ft.,  or  60  in.,  and  the  breadth  h of  the  heel  is  15  in.,  the 
number  of  the  frog  is  the  quotient  of  60  -s- 15  = 4.  Theoret- 
ically, the  length  of  the  frog  is  the  distance  from  a to  the 
middle  point  of  a line  drawn  from  k to  l\  practically,  we  take 
from  a to  l as  the  distance.  As  it  is  often  difficult  to  deter- 
mine the  exact  point  a of  the  frog,  a more  accurate  method 
of  determining  the  frog  number  is  to  measure  the  entire  length 
dl  of  the  frog  from  mouth  to  heel , and  divide  this  length  by  the 
sum  of  the  mouth  width  b and  the  heel  width  h.  The  quotient 
will  be  the  exact  number  of  the  frog. 

For  example,  if,  in  Fig.  2,  the  total  length  dl  of  the  frog  is 
7 ft.  4 in.,  or  88  in.,  and  the  width  h is  15  in.,  and  the  width  b 
of  the  mouth  is  7 in.,  then  the  frog  number  is  88  -4-  (15  + 7)  = 4. 
Frogs  are  known  by  their  numbers.  That  in  Fig.  2 is  a 
No.  4 frog. 

The  Frog  Angle.— The  frog  angle  is  the  angle  formed  by  the 
gauge  lines  of  the 
rails,  which  form 
its  tongue.  Thus, 
in  Fig.  2,  the  frog 
angle  is  the  angle 
l a'  k.  The  amount  Fig.  3. 

of  the  angle  may  be  found  as  follows:  The  tongue  and  heel  of 


314 


SURVEYING. 


the  frog  form  an  isosceles  triangle  (see  Fig.  3).  By  drawing 
a line  from  the  point  a of  the  frog  to  the  middle  point  b of 
the  heel  c d,  we  form  a right-angled  triangle,  right-angled 
at  b.  The  perpendicular  line  a b bisects  the  angle  a,  and,  by 

b c 

trigonometry,  we  have  tan  £ a = The  dimensions  of 

the  frog  point  given  in  Fig.  3 are  not  the  same  as  those  given 
in  Fig.  2,  but  their  relative  proportions  are  the  same,  viz.,  the 
length  is  four  times  the  breadth.  The  length  ab  = 4 and 
the  width  cd  — 1;  hence,  be  = £.  Substituting  these  values, 

we  have  tan  £ a = ^ = £ = .125.  Whence,  £ a = 7°  7£'  and 


a — 14°  15';  that  is,  the  angle  of  a No.  4 
frog  is  14°  15'. 

Frog  numbers  run  from  4 to  12,  inclu- 
ding half  numbers,  the  spread  of  the  frog 
increasing  as  the  number  decreases. 

The  Parts  of  a Turnout. — The 
several  parts  of  a turnout  are 
\ represented  in  Fig.  4.  The  dis- 

tance  pf  from  the  P.  C.  of  the 

turnout  curve  to  the  point  -of 

frog  is  called  the  frog  distance. 
Fig.  4.  The  radius  c o of  the  turnout 

curve,  the  frog  distance,  the 
frog  angle,  and  the  frog  number  bear  certain  relations  to  one 
another,  -which  are  expressed  by  the  following  formulas: 
Tangent  of  half  frog  angle  = gauge  -4-  frog  distance. 

Frog  number  = |/ radius  c o -f  twice  the  gauge. 

Frog  number  = 1 -r-  £ the  tangent  of  £ the  frog  angle. 

Radius  co  = twice  the  gauge  X square  of  the  frog  number. 
Radius  co  — (frog  distance  pf  + sine  of  frog  angle)  — i the 
gauge. 

Radius  co  — gauge  -j-  (1  — cosine  of  frog  angle)  — £ the 
gauge. 

Frog  distance  pf  — frog  number  X twice  the  gauge. 

Frog  distance  pf  = gauge  p q tangent  of  £ the  frog  angle. 
Frog  distance  pf  = (radius  co  + half  the  gauge)  X sine  of 
frog  angle. 


TRACKWORK. 


315 


Middle  ordinate  (approximate)  = £ the  gauge. 

Each  side  ordinate  (approximate)  = £ the  middle  ordinato 
= 1%  (or  .188)  of  the  gauge. 

Switch  length  (approximate)  = 

V throw  in  feet  X 10,000 

tan  deflection  for  chords  of  100  ft.  for  radius  co  of  turnout  curve* 
The  tangent  deflection  may  be  obtained  from  the  table  on 
pages  298-300. 

Turnouts  From  a Straight  Track. 


Gauge , 4 ft.  in.  Throw  of  switch , 5 in. 


Frog 

Number.  , 

Frog 

«5 

£ 

Turnout 

Radius. 

Degree  of 
I Turnout 

Curve. 

Frog 

Distance. 

Middle 

Ordinate. 

Side 

Ordinate. 

Stub 

Switch 

Length. 

o 

r 

Feet. 

o 

/ 

Feet. 

Feet. 

Feet. 

Feet. 

12 

4 

46 

1,356 

4 

14 

113.0 

1.177 

.883 

34 

11% 

4 

58 

1,245 

4 

36 

108.3 

1.177 

.883 

32 

11 

5 

12 

1,139 

5 

02 

103.6 

1.177 

.883 

31 

10% 

5 

28 

1,038 

5 

31. 

98.9 

1.177 

.883 

29 

10 

5 

44 

942 

6 

05 

94.2 

1.177 

.883 

28 

9% 

6 

02 

850 

6 

45 

89.5 

1.177 

.883 

27 

9 

6 

22 

763 

7 

31 

84.7 

1.177 

.883 

25 

8% 

6 

44 

680 

8 

26 

80.0 

1.177 

.883 

24 

8 

7 

10 

603 

9 

31 

75.3 

1.177 

.883 

22 

7% 

7 

38 

530 

10 

50 

70.6 

1.177 

.883 

21 

7 

8 

10 

461 

12 

27 

65.9 

1.177 

.883 

20 

6%' 

8 

48 

398 

14 

26- 

61.2 

1.177 

.883 

18 

6 

9 

32 

339 

16 

58 

56.5 

1.177 

.883 

17 

5% 

10 

24 

285 

20 

13 

51.8 

1.177 

.883 

15 

5 

11 

26 

235 

24 

32 

47.1 

1.177 

.883 

14 

4% 

12 

40 

191 

30 

24 

42.4 

1.177 

.883 

13 

4 

14 

14 

151 

38 

46 

37.7 

1.177 

.883 

11 

The  switch  lengths  in  the  above  table  merely  denote  the 
shortest  length  of  stub  switch  that  will  at  the  same  time  form 
part  of  the  turnout  curve,  and  give  .5  in.  throw.  Point  or 
split  switches  require  a throw  of  not  more  than  3£  in.,  though 
many  have  a throw  of  5 in.,  with  an  equal  space  between  the 
gauge  lines  at  the  heel.  The  heels  of  a split  switch,  which 
occupy  the  same  position  as  the  toes  of  a stub  switch,  should 


316 


SURVEYING. 


be  placed  at  the  point  where  the  tangent  deflection  or  offset 
is  5 in.  The  point  where  the  tangent  deflection  is  but  4£  in. 
will  answer  for  many  rail  sections,  but  for  those  above  65  lb. 
per  yd.,  5 in.  should  be  taken. 

In  the  table  on  pages  298-300,  tangent  deflections  for  chords 
of  100*  ft.  are  given  for  all  curves  up  to  20°;  and  for  a curve  of 

higher  degree,  the  tangent  deflection  may  be  found  by  apply- 

C2 

ing  the  formula  tan  deflection  = — 

2 K 

In  complicated  trackwork,  where  space  is  limited,  curves 
must  be  chosen  to  meet  the  existing  conditions,  and  not  with 
reference  to  particular  frog  angles,  in  which  case  the  frogs 
are  called  special  frogs  and  are  made  to  fit  the  particular 
curve  used.  The  determi- 
nation of  the  frog  distance, 
switch  length,  and  frog  angle 
may  be  understood  by  referring 
to  Fig.  5. 

Let  the  main  track  ab  be  a 
straight  line;  the  gauge  p q = 
4 ft.  8£  in.  (=  4.71  ft.);  the  de- 
gree of  the  turnout  curve  = 
13°;  the  chord  qd  = 100  ft.; 
cd  = the  tangent  deflection 
of  the  chord  q d;  and  pf  = 
the  frog  distance.  From  the 
table  on  page  299,  we  find  the 
tangent  deflection  for  a chord  100  ft.  long  of  a 13°  curve  is 
11.32  ft.  Then,  from  Fig.  5,  we  have  the  proportion 
cd:ef  = qc2  : qe2. 

Now,  in  curves  of  large  radius,  qc  and  qd  are  assumed  to 
be  equal.  Also,  qe  = pf , the  frog  distance,  and  substituting 
these  equivalents  wre  have  the  proportion 


Substituting  the  above  given  quantities  in  the  proportion, 
we  have  11.32  : 4.71  = 1002  : pf2; 

whence,  P/3  = 

and  the  frog  distance,  pf  = 64.5  ft. 


TRACKWORK. 


317 


If  the  space  between  the  gauge  lines  at  the  heels  of  a split 
switch  be  taken  at  5 in.  = .42  ft.,  the  distance  from  the  P.  C. 
of  the  turnout  curve  to  the  heel  of  the  switch  may  be  found 
as  follows: 

In  Fig.  5,  let  h,  the  tangent  offset  at  the  heel  of  the  switch 
= .42  ft.,  we  have  the  proportion 

cd  : h = qd2  : qh2, 

and  substituting  known  values,  we  have 
11.32  : .42  = 1002  : qh2, 


whence, 


qh  = 


10,000  X .42 
11.32 


= 371.02, 


and  q h = 19.26  ft. 

This  locates  the  heel  of  a split  switch  and  the  toe  of  a stub 
switch. 

The  frog  angle  is  the  angle  kfl  (see  Fig.  5)  formed  by  the 
gauge  line  of  the  main  rail  / k and  the  tangent  to  the  outer 
rail  qf  of  the  turnout  curve  at  the  point  where  the  two  rails 
intersect.  This  angle  is  equal  to  the  central  angle  qof.  The 
arcs  qf  and  r s are  assumed  to  be  of  the  same  length.  The 
turnout  curve  being  13°,  the  central  angle  for  a chord  of  1 ft. 

13  X 60 

is  — = 7.8',  and  the  central  angle  for  64.5  ft.  the  frog 


distance , is  7.8'  X 64.5  = 8°  23',  the  frog  angle  for  a 13°  curve. 
By  this  process  the  frog  distance,  switch  length,  and  frog 
angle  may  be  calculated  for  curves  of  any  radius. 

To  Lay  Out  a Turnout  From  a Curved  Main  Track. — There  are 
two  cases: 

Case  I.— When  the  two  curves  deflect  in  opposite  direc- 
tions, illustrated  in  Fig.  6. 

Case  II. — When  the  two  curves  deflect  in  the  same  direc- 
tion, illustrated  in  Fig.  7. 

In  Fig.  6,  the  curve  ab  is  3°  30',  and  it  is  proposed  to  use  a 
No.  8 frog.  By  reference  to  the  table  on  page  315,  we  find  that 
the  degree  of  curve  corresponding  to  a No.  8 frog  is  9°  31'. 
Accordingly,  we  use  a turnout  curve  a e,  whose  degree  when 
added  to  the  degree  of  curve  of  the  main  track  shall  equal  the 
degree  required  for  a No.  8 frog;  i.  e.,  we  use  a 6°  turnout 
curve,  which  is  within  1 minute  of  the  required  degree,  and 
close  enough  for  practical  purposes.  We  know  that  for 


318 


SURVEYING. 


curves  of  moderate  radii,  i.  e.,  from  1°  up  to  12°,  the  tangent 
deflections  or  offsets  increase  as  the  degree  of  the  curve. 
That  is,  the  tangent  deflection  of  a 2°,  4°,  and  6°  is  two,  four, 
and  six  times,  respectively,  that  of  a 1°  curve.  In  the  accom- 
panying cuts  illustrating  the  location  of  frogs  and  switches, 
each  curve  is  represented  by  two  lines  indicating  the  rails, 
whereas  only  the  center  lines  of  the  curves  are  run  in  on  the 
groundt  In  Fig.  6,  the  line  c d is  tangent  to  the  center  lines 
of  the  curves.  These  center  lines  do  not  appear  in  the  cut. 

Again  referring  to  Fig.  6,  if  a tangent  cd  be  drawn  at  c, 

the  point  common  to 
the  center  lines  of  the 
curves,  the  sum  of  the 
deflections  of  both 
curves  from  the  com- 
mon tangent  will  be 
equal,  in  this  case, 
to  the  tangent  deflec- 


Fig.  6. 


tion  of  a 9°  30'  curve  from  a straight  line. 

Accordingly,  to  find  the  frog  distance  for  a 6°  turnout 
curve  from  a 3°  30'  curve,  the  curves  being  in  opposite  direc- 
tions, as  shown  in  Fig.  6,  we  find  the  tangent  ^deflection  of  a 
9°  30'  curve  for  a chord  of  100  ft.  This  deflection  is  8.28  ft.,  as 
given  in  the  table  on  page  299. 

Assuming  the  gauge  of  track  to  be  standard,  viz.,  4 ft.  8£  in. 
= 4.71  ft.,  and  denoting  the  required  frog  distance  by*,  we 
have  the  following  proportion: 

8.28  : 4.71  = 1002  ; 

„ 10,000  X 4.71 


whence, 


8.28 


= 5,688.4, 


and  the  frog  distance,  * = 75.42  ft. 

We  use  the  tangent  deflection  for  a 9°  30'  curve,  which 
very  nearly  equals  the  tangent  deflection  for  a 9°  31'  curve, 
thus  saving  the  labor  of  a calculation;  this  will  not  appreci- 
ably affect  the  result. 

We  locate  the  heel  of  the  switch  in  the  same  way,  using 
for  the  second  term  of  the  proportion,  .42  ft.,  the  distance 
between  the  gauge  lines  at  the  heel,  instead  of  4.71  ft.,  the 
gauge  of  the  track. 


TRACKWORK. 


319 


In  Fig.  7,  which  comes  under  Case  II,  both  curves  deflect 
in  the  same  direction,  and  the 
rate  of  their  deflection  from 
each  other  is  equal  to  the  rate 
of  the  deflection'  of  a curve 
whose  degree  is  equal  to  the 
difference  of  the  degrees  of  the 
two  curves  from  a tangent. 

Let  the  main-track  curve 
a b be  5°,  and  the  turnout 
curve  ac  be  10°.  Then,  the 
rate  of  deflection  or  divergence 
of  the  10°  curve  from  the  5° 
curve  equals  the  divergence 
of  a (10°  — 5°)  = 5°  curve  from  a straight  track  or  tangent. 

Accordingly,  we  find,  in  the  table  on  page  298,  the  tangent 
deflection  for  a 5°  curve  for  a chord  of  100  ft.  = 4.36  ft. 
Denoting  the  required  frog  distance  by  x , we  have  the  fol- 
lowing proportion:  4.36  : 4.71  = 1002  : x 2, 


Fig.  ' 


whence, 


x2  — 


10,000  x 4.71 
4.36 


= 10,802.8, 


and  the  frog  distance,  x = 103.9  ft. 

Distances  are  not  calculated  nearer  than  to  tenths  of  a foot. 

How  to  Lay  Out  a Switch.— In  laying  out  a switch,  locate  the 
frog  so  as  to  cut  the  least  possible  number  of  rails.  Where 
there  is  some  latitude  in  the  choice  of  location,  the  P.  C.  of 
the  turnout  curve  can  be  located  so  as  to  bring  the  frog  near 
the  end  of  a rail. 

To  do  this,  take  from  the  table  on  page  315  the  frog  dis- 
tance corresponding  to  the  number  of  the  frog  to  be  used. 
Locate  approximately  the  P.  C.  of  the  turnout  curve,  and 
measure  from  it,  along  the  main-track  rail,  the  tabular  frog  dis- 
tance. If  this  brings  the  frog  point  near  the  end  of  the  rail, 
the  P.  C.  of  the  turnout  curve  may  be  moved  so  as  to  require 
the  cutting  of  but  one  main-track  rail.  Measure  the  total 
length  of  the  frog,  and  deduct  it  from  the  length  of  the  rail 
to  be  cut,  marking  with  red  chalk  on  the  flange  of  the  rail  the 
point  at  which  the  rail  is  to  be  cut.  Measure  the  width  of 
the  frog  at  the  heel,  and  calculate  the  distance  from  the  heel 


320 


SURVEYING. 


to  the  theoretical  point  of  frog.  For  example,  if  the  width 
of  the  frog  at  the  heel  is  8£  in.,  and  a No.  8 frog  is  to  be  used, 
the  theoretical  distance  from  the  heel  to  the  point  of  frog  is 
8.5  X 8 = 68  in.  = 5 ft.  8 in.  Measure  off  this  distance  from 
the  point,  marking  the  heel  of  the  frog.  This  will  locate  the 
point  of  the  frog,  which  should  be  distinctly  marked  with 
red  chalk  on  the  flange  of  the  rail.  It  is  a common  practice 
to  make  a distinct  mark  on  the  web  of  the  main-track  rail, 
directly  opposite  the  point  of  frog.  This  point  being  under 
the  head  of  the  rail,  it  is  protected  from  wear  and  the  weather. 
The  P.  C.  of  the  turnout  curve  is  then  located  by  measuring 
the  frog  distance  from  the  point  of  frog.  .From  the  table  on 
page  315,  we  find  the  frog  distance  for  a No.  8 frog  is  75.3  ft., 
and  the  switch  length,  i.  e.,  distance  from  P.  C.  of  turnout 
curve  to  heel  of  split  switch  or  toe  of  stub  switch,  is  22  ft. 

If  a stub  switch  is  to  be  laid,  make  a chalk  mark  on  both 
main-track  rails  on  a line,  marking  the  center  of  the  head- 
block.  A more  permanent  mark  is  made  with  a center  punch. 
Stretch  a cord  touching  these  marks,  and  drive  a stake  on 
each  side  of  the  track,  with  a tack  in  each.  This  line  should 
be  at  right  angles,  to  the  center  line  of  the  track,  and  the 
stakes  should  be  far  enough  from  the  track  not  to  be  dis- 
turbed when  putting  in  switch  ties.  Next,  cut  the  switch 
ties  of  proper  length;  draw  the  spikes  from  the  track  ties, 
three  or  four  at  a time,  and  remove  them  from  the  track, 
replacing  them  with  switch' ties,  and  tamping  them  securely 
in  place.  When  all  the  long  ties  are  bedded,  cut  the  main- 
track  rail  for  the  frog,  being  careful  that  the  amount  cut  off 
is  just  equal  to  the  length  of  the  frog.  If,  by  increasing  or 
decreasing  the  length  of  the  lead  5},  it  is  possible  to  avoid 
cutting  a rail,  do  not  hesitate  to  do  so,  especially  for  frogs 
above  No.  8. 

Use  full-length  rails  (30  ft.)  for  moving,  or  switch,  rails, 
and  be  careful  to  leave  a joint  of  proper  width  at  the  head- 
chair.  Spike  the  head-chairs  to  the  head-block  so  that  the 
main-track  rails  will  be  in  perfect  line.  Spike  from  8 to  11  ft. 
of  the  switch  rails  to  the  ties,  and  slide  the  cross-rods  on  to 
the  rail  flanges,  spacing  them  at  equal  intervals.  The  cross- 
rods are  placed  between  the  switch  ties,  which  should  not 


TRACKWORK. 


321 


be  more  than  15  in.  from  center  to  center  of  tie.  The  switch 
ties,  especially  those  under  the  moving  rails,  should  he  of 
sawed  oak  timber.  Southern  pine  is  a good  second  choice. 
Attach  the  connection-rod  to  the  head-rod  and  to  the  switch 
stand.  With  these  connections  made,  it  is  an  easy  matter 
to  place  the  switch  stand  so  as  to  give  the  proper  throw  of 
the  switch. 

It  is  common  practice  to  fasten  the  switch  stand  to  the 
head-block  with  track  spikes,  but  a better  fastening  is  made 
with  bolts.  The  stand  is  first  properly  placed,  and  the  holes 
marked  and  bored,  and  the  bolts  passed  through  from  the 
under  side  of  the  head-block.  This  obviates  all  danger  of 
movement  of  the  switch  stand  in  fastening,  which  is  liable 
to  occur  when  spikes  are  used,  and  insures  a perfect  throw. 

The  use  of  track  spikes  is  quite  admissible  when  holes  are 
bored  to  receive  them,  in  which  case  a half-inch  auger  should 
be  used  for  standard  track  spikes.  The  switch  stand  should, 
when  possible,  be  placed  facing  the  switch,  so  as  to  be  seen 
from  the  engineer’s  side  of  the  engine— the  right-hand  side. 

Next  stretch  a cord  from  a,  Fig.  8,  a point  on  the  outer 
main-track  rail  opposite  the  P.  C.  of  the  turnout  curve  to  br , 
the  point  of  the  frog.  This  cord  will  take  the  position  of  the 
chord  of  the  arc  of  the  outer  rail  of  the  turn- 
out curve.  Mark  the  middle  point  c and  the 
quarter  points  d and  e.  Whatever  the  degree 
of  the  turnout  curve,  the  distance  from  the 
middle  point  c of  the  chord  to  the  arc  a b'  is 
1.18  ft.,  and  the  distances  from  the  quarter 
points  d and  e are  .88  ft.;  hence,  at  c lay  off 
the  ordinate  1.18  ft.,  and  at  both  d and  e the 
ordinate  .88  ft.,  three-quarters  of  the  middle 
ordinate.  These  offsets  will  mark  the  gauge 
line  of  the  rail  a b' . Add  to  these  offsets  the 
distance  from  the  gauge  line  to  outside  of  the 
rail  flange,  and  mark  the  points  on  the  switch  ties.  Spike  a 
lead  rail  to  these  marks,  and  place  the  other  at  easy  track 
gauge  from  it.  Spike  the  rails  of  the  turnout  as  far  as  the 
point  of  frog  to  exact  gauge,  unless  the  gauge  has  been 
widened  owing  to  the  sharpness  of  the  curve.  Beyond  the 


Fig.  8. 


322 


SURVEYING. 


point  of  frog  the  curve  may  be  allowed  to  vary  a little  in 
gauge  to  prevent  a kink  showing  opposite  the  frog.  In  case 
the  gauge  is  widened  at  the  frog,  increase  the  guard-rail  dis- 
tance an  equal  amount.  For  a gauge  of  4 ft.  8£  in.,  place  the 
side  of  the  guard  rail  that  comes  in  contact  with  the  car 
wheels  at  4 ft.  6£  in.  from  the  gauge  line  of  the  frog.  This 
gives  a space  of  1 $ in.  between  the  main  and  guard  rails. 

In  case  the  gauge  is  widened  i or  | in.,  increase  the  guard- 
rail distance  an  equal  amount. 

When  the  turnout  curve  is  very  sharp,  it  will  be  necessary 
to  curve  the  switch  rails,  to  avoid  an  angle  at  the  head-block. 
The  lead  rails  should  be  carefully  curved  before  being  laid, 
and  great  pains  should  be  taken  to  secure  a perfect  line. 

If  a point , or  split,  switch  is  to  be  laid,  the  order  of  work  is 
nearly  the  same.  The  same  precautions  must  be  taken  to 
avoid  the  unnecessary  cutting  of  rails,  with  the  additional 
precaution  of  keeping  the  switch  points  clear  of  rail  joints, 
as  the  bolts  and  angle  splices  will  prevent  the  switch  points 
from  lying  close  to  the  stock  rails.  As  already  stated,  these 
conditions  can  usually  be  met  where  there  is  some  range  in 
the  choice  of  the  location  of  the  switch.  Where  there  is 
none,  the  main-track  rails  must  be  cut  to  fit  the  switch. 

Having  located  the  point  of  frog,  the  P.  C.  of  the  turnout 
curve,  and  the  heel  line  of  the  switch,  measure  back  from 
the  heel  line  a distance  equal  to  the  length  of  the  switch 
rails,  and  place  on  the  flange  of  each  rail  a chalk  mark  to 
locate  the  ends  of  the  switch  points.  This  will  also  locate  the 
head-block.  Prepare  switch  ties  of  the  requisite  number 
and  length,  and  place  them  in  the  track  in  proper  order.  As 
in  the  case  of  stub  switches,  see  to  it  that  all  long  switch  ties 
are  in  place  before  cutting  the  rail  for  placing  the  frog; 
also,  that  the  ends  of  the  lead  rails,  with  which  the  switch 
points  connect,  are  exactly  even;  otherwise,  the  switch  rods 
will  be  skewed,  and  the  switch  will  not  work  or  fit  well. 
Fasten  the  switch  rods  in  place,  being  careful  to  place  them 
in  their  proper  order,  the  head-rod  being  No.  1.  Each  rod  is 
marked  with  a center  punch,  the  number  of  the  punch 
marks  corresponding  to  the  number  of  the  rod. 

Couple  the  switch  points  with  the  lead  rails,  and  place  the 


TRACKWORK. 


323 


sliding  plates  in  position,  securely  spiking  them  to  the  ties. 
Connect  the  head-rod  with  the  switch  stand,  and  close  the 
switch,  giving  a clear  main  track. 

Adjust  the  stand  for  this  position  of  the  switch,  and  bolt  it 
fast  to  the  head-block.  Next,  crowd  the  stock  rail  against 
the  switch  point  so  as  to  insure  a close  fit,  and  secure  it  in 
place  with  a rail  brace  at  each  tie;  then  continue  the  laying 
of  the  rails  of  the  turnout. 

If  there  is  no  engineer  to  lay  out  the  center  line  of  the 
turnout,  the  section  foreman  can  put  in 
the  lead  from  ordinates,  as  explained  in 
Fig.  8.  In  modern  railroad  practice,  how- 
ever, most  trackwork  is  done  under  the 
direction  of  an  engineer,  in  which  case 
the  center  line  of  the  turnout  is  located 
with  a transit.  This  insures  a correct  line 
and  expedites  work.  For  ordinary  curves, 
center  stakes  at  intervals  of  50  ft.  are 
sufficient,  excepting  between  the  P.  C.  of 
the  turnout  and  the  point  of  frog,  where 
there  should  be  a center  stake  at  each 
interval  of  25  ft.  Place  a guard  rail  oppo- 
site the  point  of  frog  on  both  main  track 
and  turnout.  The  guard  rail  should  be 
10  ft.  in  length;  this  is  an  economical 
length  for  cutting  rails,  as  each  full-length  rail  makes  three 
guard  rails. 

Two  styles  of  guard  rails  are  shown  in  Fig.  9.  That  shown 
at  B is  in  general  use,  but  the  style  shown  at  A is  growing  in 
favor.  The  latter  is  curved  throughout  its  entire  length.  At 
its  middle  point  a,  directly  opposite  the  point  of  frog,  the 
guard  rail  is  spaced  If  in.  from  the  gauge  line  of  the  turnout 
rail  be.  From  this  point  the  guard  rail  diverges  in  both 
directions,  giving  at  each  end  a flangeway  of  4 in.  This 
allow;s  the  wheels  full  play,  excepting  at  the  point  of  frog, 
where  the  guard  rail  is  exactly  adjusted  to  the  track  gauge, 
and  holds  the  wheels  in  true  line,  preventing  them  from 
climbing , or  mounting , the  frog.  The  style  of  guard  rail  shown 
at  B,  though  still  much  used,  has  two  objectionable  features; 


Fig 


324 


SURVEYING. 


viz.,  first,  the  abruptly  curved  ends  d and  e often  receive  an 
almost  direct  blow  from  the  wheel  flanges,  which  causes  a 
car  to  lurch  violently;  and  second,  the  flangeway  of  uniform 
width,  though  proper  for  the  main  track  when  straight,  as  in 
Fig.  9,  is  unsuited  for  sharp  curves  on  either  a main  track  or 
a turnout,  as  it  compels  the  wheels  to  follow  a curved  line; 
whereas  the  normal  position  of  the  wheel  base  of  each  truck 
is  that  of  a chord  of,  or  a tangent  to,  the  curve.  These  two 
defects  alone  produce  what  is  known  as  a rough-riding  frog, 
even  though  the  frog  is  well  lined  and  ballasted. 

Location  of  Crotch  Frog.— A crotch , or  middle , frog  is  a frog 
placed  at  the  point  where  the  outer  rails  of  both  turnouts  of 

a three-throw  switch 
cross  each  other. 
When  both  turnouts 
are  of  the  same  de- 
gree, the  crotch  frog 
comes  midway  be 
tween  the  main-track 
rails.  Its  location 
and  angle  may  be  determined  as  follows:  Let  the  turnout 
curves  A and  B,  Fig.  10,  be  each  9°  30',  uniting  with  the  main 
track  C by  a three-throw  switch.  Let  a be  the  P.  C.  common 
to  both  curves,  and  b,  the  location  of  crotch,  or  middle,  frog. 

It  is  evident  that  the  point  of  the  crotch  frog  should  be 
exactly  midway  between  the  gauge  lines  of  the  main-track 
rails,  and  if  the  gauge  is  4 ft.  8&  in.  = 4.71  ft.,  the  point  of 


the  crotch  of  the  frog  will  be 


4.71 


= 2.35  ft.  from  each  rail. 


Now,  the  problem  is  to  find  the  frog  distance  from  a,  the 
P.  C.,  to  the  point  c,  where  the  tangent  deflection  will  equal 
2.35,  or  half  the  gauge.  From  the  table  on  page  299,  we  find 
the  tangent  deflection  of  a 9°  30'  curve  is  8.28  ft.  Applying 
the  principle  explained  in  connection  with  Fig.  5,  and  letting 
x represent  the  required  frog  distance,  we  have  the  following 
proportion:  8.28:2.35  = 1002 : x-; 


whence, 


x2  = 


1002  x 2.35, 
8.28 


2,838.2  ft., 


and  the  required  frog  distance  x = 53.3  feet,  nearly. 


TRACKWORK. 


325 


Now,  there  are  two  curves  starting  at  the  common  point  a; 
the  outer  rails  intersect  at  5,  and  the  angle  d b e,  formed  by 
tangents  drawn  at  the  point  of  intersection,  is  the  angle  of 
the  crotch,  or  middle  frog.  The  angle  is  equal  to  the  sum  of 
the  angles  afb  and  af'  b;  that  is,  equal  to  double  the  central 
angle  of  either  curve  between  the  P.  C.  and  the  point  of 
intersection  b.  The  degree  of  the  curve  is  9°  30'  = 570',  and 

570' 

the  central  angle  or  total  deflection  for  each  foot  is  — 

= 5.7';  and  for  the  frog  distance  of  53.3  ft.,  the  central  angle 
is  53.3  X 5.7'  = 303.8'  = 5°  03.8'.  The  angle  of  the  crotch  frog 
is  double  this  angle;  i.  e.,  5°  03.8'  X 2 = 10°  07.6'.  The  crotch 
frog  should  be  accurately  located  and  spiked  in  place  before 
the  lead  rails  are  placed. 

The  one  objection  to  the  three-throw  switch  is  the  open 
joint  at  the  head-block,  the  inevitable  attendant  of  the  stub 
switch,  but  its  advantages  are  so  great  that  it  will  continue  to 
be  used,  especially  in  yard  service. 

Crossover  Tracks.— A crossover  is  a track  by  means  of  which 
a train  passes  from  one  track  to  another.  The  tracks  united 
are  usually  parallel,  as  are  the  tracks  of  a double-track  road. 
Such  a crossover  is  shown  in  Fig.  11.  The  tracks  a b and  c d 
are  13  ft.  apart  from  center  to  center,  which  is  the  standard 
distance  for  double  tracks.  The  crossover  consists  of  two 


Fig.  11. 


turnout  curves,  ef  and  g h.  These  curves  are  usually,  though 
not  necessarily,  of  the  same  degree.  The  curves  terminate 
at  the  points  of  frog / and  h,  between  which  the  track  fh  is  a 
tangent.  The  essential  point  in  laying  out  a crossover  is  to 
so  place  the  frogs  that  the  connecting  track  shall  be  tangent 
to  both  curves.  In  Fig.  11,  suppose  the  frogs  are  No.  9, 
requiring  7°  31'  turnout  curves. 

From  the  table  on  page  315,  we  find  the  required  frog  dis- 
tance is  84.7  ft.,  and  the  switch  length  25  ft.  As  previously 


326 


SURVEYING. 


noted,  if  there  is  considerable  range  in  choice  of  location, 
the  frogs  can  be  so  placed  as  to  largely  avoid  the  cutting 
of  rails;  but  usually  crossovers  are  required  at  certain  precise 
places,  and  the  rails  must  be  cut  as  occasion  demands.  Hav- 
ing located  the  point  of  frog  at  /,  we  determine  the  point  of 
the  next  frog  at  A,  as  follows:  A No.  9 frog  is  one  that  spreads 
1 in.  in  width  to  every  9 in.  in  length;  and,  as  the  track 
between  the  frog  points  is  straight,  the  distance  /A  between 
these  points  will  be  as  many  times  9 in.  as  is  the  space  k 
between  the  tracks  at  the  frog  point  /.  The  main-track 
centers  are  13  ft.  apart,  making  the  space  between  the  gauge 
lines  of  the  inside  rails  8 ft.  in.  As  it  is  the  rail  l of  the 
turnout  that  joins  the  second  frog  at  A,  we  subtract  the  gauge, 
4 ft.  8*  in.  from  8 ft.  3£  in.,  leaving  3 ft.  7 in.,  the  distance 
k,  between  the  gauge  line  of  the  rail  l , opposite  the  frog 
point  /,  and  the  gauge  line  of  the  nearest  rail  of  the  track 
c d.  This  distance  multiplied  by  9 in.  will  give  the  distance 
from  the  frog  point  / to  the  frog  point  A;  3 ft.  7 in.  = 43  in.; 
43  X 9 = 387  in.  = 32  ft.  3 in.  Accordingly,  having  located 
the  point  or  frog  /,  we  mark  a corresponding  point  on  the 
nearest  rail  of  the  opposite  track.  From  this  point  we  meas- 
ure along  the  rail  the  distance  32  ft.  3 in.,  locating  the  second 
frog  point  A,  and  again  the  frog  distance  84.7  ft.  to  the  P.  C. 
of  the  second  turnout  curve  at  g. 

If  frogs  of  different  numbers,  say  7 and  9,  were  to  be  used, 
the  distance  between  the  frogs  is  found  as  follows: 

As  the  No.  7 frog  spreads  1 in.  in  7 in.,  and  the  No.  9 frog  1 
in.  in  9 in.,  the  two  will  together  spread  2 in.  in  7 4-  9 = 16  in., 
or  1 in.  in  8 in.  Now,  if  the  rails  to  be  united  are  3 ft.  7 in., 
or  43  in.,  apart,  as  in  the  previous  problem,  the  distance 
between  the  frog  points  will  be  43  X 8 = 344  in.  = 28  ft.  8 in. 

In  locating  crossover  tracks,  regard  should  be  paid  to  the 
direction  in  which  the  bulk  of  the  traffic  moves,  and  the 
crossover  tracks  should  be  so  placed  that  loaded  cars  will  be 
backed,  not  pushed,  from  one  track  to  the  other. 

At  all  stations  on  double-track  roads  there  should  be  a 
crossover  to  facilitate  the  exchange  of  cars  and  the  making 
up  of  trains. 


PERPETUAL  CALENDAR. 


327 


PERPETUAL  CALEN  DA R— 1797-1904. 


3 

4 

5 

6 

0 

1 

2 

June. 

Sept. 

Dec. 

April. 

July. 

Jan. 

Oct. 

May. 

Aug. 

Feb. 

Mar. 

Nov. 

1797 

1798 

1799 

1800 

1801 

1802 

1803 

1804 

1805 

1806 

1807 

1808 

1809 

1810 

1811 

1812 

1813 

1814 

1815 

1816 

1817 

1818 

1819 

1820 

1821 

1822 

1823 

1824 

1825 

1826 

1827 

1828 

1829 

1830 

1831 

1832 

1833 

1834 

1835 

1836 

1837 

1838 

1839 

1840 

1841 

1842 

1843 

1844 

1845 

1846 

1847 

1848 

1849 

1850 

1851 

1852 

1853 

1854 

1855 

1856 

1857 

1858 

1859 

1860 

1861 

1862 

1863 

1864 

1865 

1866 

1867 

1868 

1869 

1870 

1871 

1872 

1873 

1874 

1875 

1876 

1877 

1878 

1879 

1880 

1881 

1882 

1883 

1884  . 

1885 

1886 

1887 

1888 

1889 

1890 

1891 

1892 

1893 

1894 

1895 

1896 

1897 

1898 

1899 

1900 

1901 

1902 

1903 

1904 

Sun. 

Mon. 

Tues. 

Wed. 

Thur. 

Fri. 

Sat. 

1 

2 

3 

4 

5 

6 

<7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

.34 

35 

36 

37 

38 

39 

40 

41  / 

42 

43 

44 

328 


PERPETUAL  CALENDAR. 


By  means  of  the  table  given  on  the  preceding  page,  the  day 
of  the  week  corresponding  to  any  date  between  1797  and  1904, 
inclusive,  may  be  readily  found.  Before  every  leap  year 
there  is  a blank  space.  To  find  the  day  of  the  week  on 
which  January  1 of  any  year  fell,  find  that  year  in  the  table; 
glance  down  the  column  containing  that  year,  and  the  day 
of  the  week  at  the  foot  of  the  column  will  be  the  day  of  the 
week  required.  Thus,  to  find  on  what  day  of  the  week 
January  1, 1895,  fell,  we  find  under  1895  in  the  table,  Tuesday. 
For  leap  years,  we  look  for  day  of  week  under  the  blank 
space  before  the  year.  Thus,  January  1, 1896,  fell  on  Wednes- 
day, Wednesday  being  in  the  column  containing  the  blank 
space  before  1896.  To  find  the  day  of  the  week  for  any  other 
date,  add  (mentally)  to  the  day  of  the  month  the  first  number 
under  the  day  of  the  week  that  is  contained  in  the  column 
containing  the  year  of  the  century;  to  this  sum,  add  the 
number  above  the  month  at  the  top  of  the  table.  Find  the 
number  thus  obtained  in  the  columns  of  figures  under 
the  days  of  the  week;  the  day  of  the  week  at  the  head  of  the 
column  containing  this  number  will  be  the  day  required. 
Thus,  to  find  on  what  day  of  the  week  September  10, 1813, 
fell,  we  find  1813  in  the  table.  The  number  under  the  day  of 
the  week  in  the  column  containing  1813  is  6,  and  the  number 
above  September  at  the  top  of  the  table  is  4.  Hence, 
10  + 6 + 4 = 20.  The  day  of  the  week  above  20,  in  the  lower 
part  of  the  table,  is  Friday. 

For  dates  in  January  and  February  of  leap  years,  take  one 
day  less,  or  add  the  number  beneath  the  day  of  the  wTeek 
under  the  blank  space  preceding  the  year.  Thus,  for  Feb- 
ruary 12, 1896,  we  have  12  + 4 + 2 = 18,  and  the  day  of  the 
week  above  18  is  Wednesday. 

The  table  may  also  be  used  for  fixing  dates.  Thus, 
Thanksgiving  Day  is  the  last  Thursday  in  November;  on 
what  day  of  the  month  did  it  fall  in  1897  ? Since  the  earliest 
day  on  which  it  can  fall  is  the  24th,  we  find  on  what  day  of 
the  week  November  24  falls,  and  then  count  ahead  to  Thurs- 
day. Referring  to  the  table,  24  + 6 + 2 = 32;  the  day  of  the 
week  above  32  is  Wednesday,  and  since  Thursday  is  one  day 
late*,  it  follows  that  Thanksgiving  Day  in  1897  fell  on  the  25th. 


